
Coppght]^". 



CDFYRICHT DEFOSm 



FRESHMAN CALCULUS 

A PffeSENTATION OF 

Fundamental Conceptions and Methods 

FOR 

Students of Science and Fngineering 



BY 

WiaiAM R Ransom, AM. 
Professor erf Molhmaiics inTuffs Col/e^e 



Uthographed LomoN, 






LIBRARY of CONGRESS 

Two Oi:Di?s^ Received 

JAN la 1909 

Oopyritfnt Entry 

WASS SU XXc, NO. 

COPY a. 



Copi/ricjht, /909. 
Bij WWhm R. Ran&orrt, 



Preface. 

The purpose of Freshman Cahuiub 
is to provide the student of science of 
engineering vQrq eatltf in his course 

viith a fQmiliQritif with the fundi ameh- 
to I conceptions and methods of the 
calculus in gs far as the if are of ase 
in the elementurif studtf of the pht^sh 
cqI sciences, Onlij the chief topics of 
the convent tonat caiculus course Qre 
touched upon: rates and sums , UhJh, 

maxima and minima, expansions, 'the 

interpretation of slopes and are<is incase 
of the graphs of pht^sical functions, 

a nd differ en tiation and in terqra Hon of the 

simplest and most usual forms. 

An abundance of concrete prot>lew% 
not too far fetched , hov^ been provided 
in which reosonabie data are qiven 
and specific /jumerical results arere- 
quired, tic problems have been intro- 
duced which ore to be solved by 
substituting^ into Q formula. Great em- 
phasis IS /aid upon the careful use 
of concrete numbers, the proper umh 
bein^ never omitted. 



In ttie f-heon^ pr»sente(f, simplicif^tf 
&nd dnectneas have been soucfht, nev^r, 
(the cfuther hel/e/es) at the expense oftm- 
CMract^, The dm cuss ion is J /mi ted to 

functions whose cf rap h^ are the fa m]^- 

iOf smooth curves of element arif ph(j^ 

SIcqI science: a direct appeal is made 

to intuition ot manif, points where a 

ri(f era us analtftic at demonstration 

would he not convincing , nor even in- 
telligible to I- he in) mature student 

Ifle author wishes to acknowledge 

the inspiration he has received tronjFto- 
fessors W.t\ Osgood, of Mar yard Umyer- 
sitHf Itvinq Fisher, ot YQ/e Universiti/^antl 
D.r, Campbell, of Armour Jnslitute of 
Technology, 'whose recent treatises on the 
Calculus have encouraofed him to under- 
take the tasK of prepurinq a simple but 
thorough text for the use of^ first if ear 
students) and to express fyis than lis to 
Mr, A. Dillingham, of Tufts College,for 
his- re r if material assistance With the 
problems. 

Tufts College f^ass. 
December, nOQ, 



Contents 

VartabJes and f^unchons. A: /-s pa^B & 

dm its, y4;^7 -f 

Piol-ph'on PtololemSj A d-zZ h 

IncrementSj B: i-tz 8 

Difference QuoHehts, Bi f3'is to 

Det-iVah'yeSj 3: f(?-2 6 iz 

Several Dependent Var/ables, tS 

Limits of Products otnd Quotients, Ih 

Differerrha/s, tj 

iffe rerttfal f^orrn uIq%, / 8 
Oerivah'yes bi/ Dffferenticf/ Formu/as, 2Z 

Drift on Diffor&r}tiafion^ C: l-t^ ^Y 

Successive Differentiation, C: 25^35 zb 

Suhstitufion into O^rircrhves, C: sh-^q z8 

Shapes of Curves, C:^o-4-S 30 

EXGTcr Pates of in c recuse, sz 

D e riven tive pa te Problems, D .* / - ^ 3 sf 

Maxima and M-tnima, 5l 

MQx'inr\um-Mihimuiyi Problem % £:i-'4-o 68 

linte^ration, (>B 

Drill on Integrrafio/i, P: 1-22. 'jz 

Deterrniha/^i'on of Constant, P:z3-2i 74 

Integral Pate Pro blends, ^' f-2^ 7^ 



TranacendenM Functions, 64 

t>nH on Wanscendenhals , J^: l-zs <fo 

M'iscGllan&ous Problems^ I: /-5/ 9^ 

D e ffnite Inte^raJs^ J: 1-10 10 z 

Areas, Hi i-/s loj 
inl'^rprehatfon (?f Slopes Sfid Area% H-iy-za H4- 

ScalQr Slopes Qnd Areas, h:zi-3o tra 

Loiv of- {"he Meon, t2z 

Tci(//or-s Expansion^ L: l-zo iz4 

Tacf/or's Forma/Q Ex^e/icfeai, L: zi-z^ 130 

Approx/matm fc>rmu/a8, L: jiy-30 fsz 

Small Corrections, L: 31 -3z /33 

Absolute Errors^ L: d^-^i iss 

Re/cft/Ve Errors^ L:az-^6 aa 

Integra tf on Qs Sunnmottion, )^o 

Summah'on Problems, All I-10 /4^ 

dumm at/on ^l&menls. Mi n-ih /-^^ 

In testation bif pQrts, N: i-to }5o 

fmdmg Lists^. ^sz 

TQbles, Radians and Q-Logarithm^. /54 

place in the Cmrnculumj i5b 

Spec tail u N cimed P cites, iB& 



FRESHMAN CALCULUS 



In the sf■^o/Lf of- arithmetic, algebra. 
Qhcf thgonomehr^fi the student has loeen 
concerned chiefli^ ivith the fihdinq of 
the numerical voilues of unhnoiA^n 
quQntf't/es, In analtfh'ccrl ^eornetrt^ 
one. begins to thfy?/r of va ri'cilole.% 
and to recognize tJiah the nnotnner. 
ih which a c^uayitftij yotries maij be of 
as yr^Qt importance as partiti^Joir 
valuer assumed under poirt/cuJar 
Condi tions. In scientific worir one 
is concern&a vertf lar^Qlcf with c^uan" 
hh'es which varCfy and whose nnanner 
of' ranbitibn is a con si deration of 
prfmarcf im porta nee ^ For the treat" 
rnent or such problems ^ knowie d^e- 
of the Calculus is requf&lte. 

The Calculus is that hroinch of t^ath- 
eionat/cs which treats of prohi^rns in 
which an ess<^nti'al element is the 
yonation of (Quantities fn valued. 



y^RJABLES AND FUNCTIONS 

T^o (fuant'fh'es are called Variables fC 
the value of one depend 3 upon the- 
voilue of the ohhen One is 00 //ed the 
Indepshd^nt variable^ or AnauMtNT; the. 
0EPEN0BN7 yarioibl^ is called a Function 
b^ hhe othen The sf-atemenf' that u is 
Of fun ct- tan of dc is thus Qbhreyial-ed: 
us. FC<) or Lj^ ^ix) e/c. 

A special nneanmof tno^j b^ ^fven to 

F( ) or ' a^i 1 etc, 

30 that either shall denote yy/joit op-er- 
ahans are to be performed upon the 
argument to produce the number u^ 

Thus if ^(r-)^ Z-xJ'-tlogic 
the sj/mbol H ) /s made to mearr/'^S^tiioir^ 
the argument, double //f and adid its lo^arii-Hm". 
Soj for KOfr/ods other ar^uments^ we have: 
fCz) 5= 2.7^ ■+ log 2: 
f(i ) ^ Zx!^ -*■ Zo9 I -2. 

Funoh'ons maj/ haf/e seyerdl oir^u merits. If 
(p(:fc,y,Z)&^^yz\ then q{(X,z,t)- i-x/ etc, 
Ar\ Qive three concrete cases of related 
variables, Sau in e a en cctse ivh'itih tjou. 
naturally t/iink of as depending on -which. 



There are f'hrffe prihcipGl woti^s of^ express- 
fri^ retatton b^h^^en fanohdfi <sincf ar^ment; 

/. By a FoFiMuLA (an t^^u^toN, or a Rulb) 

Z. Bif a(SRAPH 

^, Bjf Q TABi.£ 

MoMENCL>4n/Re 



Qenerai/ 



Tabies 



Grcrptiical 



Formulos 



Inosp. Vah, 



ARdUMENT 



tAARGiN of Table 



lioRUONTAL DtSt., 

Hun, Abscissa 



Pep, Vab, 



Function 



Bopy of Tabi e 



Vi'RTICAL DfSt., 
Rise, Ordinate 



Uterai quotihty i^lnose ualoie /$ 



assumed 



iCj Xj ^, :,._, €,fc . 



cafcc4loted 



Solved out 



Symbols 



tf, fC%0,a)Cf),. 



A-Z If F( )s '^Si^uare t-he argument, triple, 
the result^ add ^ and tt>en multipljf t)y the 
oirsumenK what Is Fcic)! fio)? Fin? rc-y.^P, 

y$-3 IF i(z)^yl:g:C7,-z), pht the curve 6^= f (^5, 

h4 If ^i-y!)^ sin['%r^^)jabbttate cjpf-Jic) 
for 'X- values trom otolj int&rvah of yg. 

A-5 If FC^O^ -x^-a^xf+ax, iind }=ixff{),and 
find ^ [F(x-fi{) - Flxi'}^ 



UMtTS 



This sufmbol L fcic) 

means Me liMir approached by fC ) 
when its ot^uinent approaches the 
value a as Ifmit. if the ^rapln at 

IS continuous J that is unbrofren, at 
'X'^cuy this Limit yt^ill be e^u&l to 

f CCL) 

anc^ rnay he found btf direct substitution. 

The only ^ort of dtscontinuity occurring in 
eleme/itG/n/ ca/cu/us is that w/iich in- 
volres Q yQni^hiny denominotan A Func- 
tion m froichi'onail form moty approach a 
limit when its argument approaches Ci 
certain number, ctnd ji^et this, limitnotbt 
ascertciincible bjf substituhdn in case this 
results in o zero denominoitor. This is 
a very common Qnd important coise in 
the calculus. In zuch cases some hroins- 
formoition in the form of the function 
must be macfe before passtn^ to the limit. 



sr 

7)v(? limits af^ ^reqt tmporhanee In A/?*. 
Calculus^ but" of- special di^h'cultij^ ate. 

The value of- the hrst of tliese depends 
on tiie Units in which fi is measured, 
Ta/re n ^ //7«f number of- anils w the 
p^RiGON (Beo"), Then tahe a smali arc 
at a circle I draw its chore! and two 
tangents. Fut the angle s 8 uhitsi 
\\ half cfiordK half-arc < one tangent 
1/ * or: r simK-K ^^ < r tan % 
or: 1 <(iu -rn)i-(sin HitH sec ^ 
As fi^o^ the oute r Urn it ten ds to ward 1, 
so the h'hjit of the included vari- 
able must be Oi>te, Then the h'rnit of 
its diWsor nnust he the constant, (Zn-r'n) 

When fi is measured in Degrees, ns-S^o 

^.fT-fn = ^.^a 32... 4-3^0 =.0i«j45'f 
When Radians are used this limit is On-e^, 

J* [(1-*-^)^ J-2.7/aBd+,co/lec/5e. A proof 
of this is too diftttult fc/r an elementarc^tejif, 



Notation Problems 

AS Express in the three n^acfs the tunc- 
tfonal relation def't^een the number ot 
seconds & hQcKtf fias been fo/h'n^ qrrcf 
fhe tiutriber of- fe^t if tias fallen. 

/4-9 In fhe expression L=IoqCn) is \h€%(jm- 
ho\ JoqC ) compeirat3le with a stfmbol 
like fc ) 7 What operahi'on does FC ) 
direct one t-o perpornt i-f Fw-^log^^XI 

A' 10 A^ tv/?<:it i/'alues of %. is the func- 
tion q> fW ^ ;f ^j : dfsc ontinuou s ? 

A-U Express-, jbcf a vetbai rule, hh^mean- 
,ln^ of FC •), If FCt)S: (x^-^^0^. 

A-lfi Is there ancf dtBconhnuitij in fhe 
graph of f(>^)^(K.^i)-riX'i)?, Does this 
function gfve qU the values of fcic) just 
as fhecf are ^iH^en bjf this .fCfQ^ ^+it 

/\-/4 If FCx)= (^)^ fabu/ate fC^) for 



7 
■x==j/a, 3, »o, loo; Does it tend hv/ard 6? 

v4-i5 G'iven <^(%)^ ^' z ^ ', Compare Me 
valuer of cpCz^x) and ^ rq>(*07^-l. 

when f(x; = ^^ Ans. jx^+sxA +fi"* 

/4-/7 //^ the relQHom between distance, d, 
moved /n t-frne^^^ fs ^ik^err bjj the hr- 
mulof <i=^fCt), wtfi<3it concrete c^uantiti^ 
fs denoted bcj Tf ^^^t,) - -FC tj] ? 

A-i9 tf f(xi^Zx?'^l, h'nd i [fcic+fi) - ^d^J 

A'^0 Giv<E>, cp(fi)-0-^A}''^ Phi- xj-cpm 
for f'he values -^=^,1, .5", .l,.oooi 
Ooes the ^raph hnd towQi^d \^-o, ^«^J? 

A-Z\ If cy^FCx) iS f-tie ec^uation of o cer- 
tain ordpfi, 3hoh/ t)U diagram that a chord 
betiveen points iVncre 70 is a* dnd a.-^t 

hois [Fra-^ei-Fca)] -re for its slope,' 



8 

iNCfiBMENTS 

The pn'moiri/ Quosh'ojn abocif' i^arioble^ /S: 
a/- what- Rates do thej/ iNCREAsst 
The ACT(/AL INC/fEASE fh Q c^u&htJttf /s 
denot&d iby writing o A (defta) before 
fhe Sijmbol for th^ quantify. When Deltas 
ate used before related variables thet/ 
rpean Corresponding iNCREMSihrs. Thus 
ifh )s the temperature at h o'- 
clock, At means a lapse of time (w 
hours) and Zl ^ means the r/se in 
temperature (ih °'s) during that ihttr- 
yal, Ai\* mai^ come out a necfoitive. 
number and thus represent q decrease. 

The increase produced in Fl-^) bj/ 
qivinq x- an increment, A-x, will be. 

AQ, ]$ a single number^ like the stf/v- 
bols sin 9, lo^N, f(%),ek^ Nor cr product, 

£xa/nple:tf Q & Cl-\'^) fihd AQ.. 
SoJatloni Q-t-AQ = ((- [t+A^]*") 

Subtracting: A<k = - 7.AXi\-1*l*iA^) 



9 

3-1 If FW2 a*»* find FC-x-^-d-*:)-Fec) 

3't If f(%)f^ shoiv thQl A^OL)^ v^wt^^x 

&-3 Ifu^sinx, find Ay when x-^%o''a/icl Ax^l' 

5-4 /f y = fC*:) be plotted J show by dio^rarrt 
that the slope of a cethoin chord I's -^. 

D-5 What does s^ py^, '"^ Jtv A77//es /s 
7ny distance from town at f» mm. past 5? 

B"7 //O -^, sttow thatACX^^ ~^^ V 

B-<S //^ Q --xHT ^ork out ^"^. J ffl 

B-9 If Q^xtr , X'^/j^a^C'^.olV ^ "^ 

B-/^ TA^/7S <?f ea/*/-/? ore duQ hi/ tke end 
of W iveehs. Meaning of T-i-WC aT-^^w^. 

B-H If u^loQ^sin{cJ*X*X show that ^^ 



jgv+A^ 



B-12 R=Vo^^; frove that -fS; ^ - ^rh-^r 



-10 

OfFFeFfENCE QuOTfENTS 

Many important' re I oh' on s between pht/icol 
(^UQnh'tCeB occur in the torm Fi ^ S-t T as 
Speed ^. Distance ~ Tfm€ 
D&nS/tjf ^ Moss -r Volume 

Acceleration -^ 3p&ed -r- Tim^ 
Slope « fprse -r Run 

Mileage "^ Far-e 4 Otstanee 

In these and similoir eases th^ last ti/t^o 
must l;)e Corr^spon dinq amounts^ so 
th e lNCR£Me NT NoTAJtoN ]s convenien t, 
Sp eed ^ A (distancej-r Ait/hie ) efc. 

Thus if distance "s^ mi, and time^^Tir, 
AX'^f Speedy fn mi. per hr.j duriyi^ tine 
^^'Xthne- interval denoted bif Ati. 
T/jiS need not be ec^ual to jt^-rt, tor 
cor) side r a case />? yirh/ch ttie Journetf 
^. ,..•' d/'d not hejin until f had 

t p^^''^ reached the value 1 hr, 



^* Moreover the abore tor- 

^ mula holds true only if 
the speed is Constant. 
If ttie speed vanes m ttie inter ml A^ 

/s the AvEFTAGEjOr Mean, Speed during Al, 



Si'm'flarljf, the Avg, or Mean densltij o^ ci 
piece ot mot-erfcil whose voluime is AV 
CLi. c/n ofnd Wfio^e mcuss is Arrv grows /s 

[Am-r A^l ^rams per cu.cm. 
The roitfo of two correspondi'yi^ increments 
fs colled Q OtFF^Rnuct QvoT/eNT. ^yerjf 
Cfyera^e reitej e^eri^ quantitjj measure ct 
m u flit's nof^eof with a '*per*' macf be 
expressed bif a DiF. Quo^ 

If two c^uanHhes are connected bj^ 

a f-ormuloi , ^ , ^ 
y = FCx) 

Exctwple: Find Mean Speed hr onif mthr- 
valy of bQ/I pQ/Iin^ /Sl^ ft, m i seconds. 
Soluhan: Put distances -^ = tL>%^ 

A-K.^ /^(^.^A^)'*-1fc>t^= ^^A\^\t,A^ 

^ - (3£ -k- \lo brO^ fi.per Sec,, mean durrn^ A% 

B~/3 f^fnd the Dif. Qvo. for ^~S7c?-z-K-h\. 

B-14 Ftnd the mean acceleration for anif 
fnteri^cfl if the speed is ^^ ft, p, sec, 
at the end of i sec, 

h-\5 f^or a <^rQph, ^ is on Avq. What^ 



DeRIVATI\/ES 



J//7C€ A^ /s /he (C^c/ua/ /hcnfase /h -c 
proc/ucec^ >V ^^ />?creasejAX, //7Xj A 
D/f. Quo.^ Ay-irA%^a^e/?o7^^ 7)ieAyBH' 
AQ,^, ^/'Mean, ra/e <?f /hcrtfase /h y 
as o(^n7/parei^ w///? x /h /h^ /h/eri/a^/ 
a^e/7cf/et:/ Ay the A. /f /s /?7easc/re^/n 
i/p/fs of y,P^^ z^/?// {?fx^ , Thus lY ^ /s 
ih Td/js, X //? m/'/es, ^^ /s /h Tons per mi. 
l/l//7(e/7 £p yery sr77(?// /n/er^a/ /s cons/Wfrnl 
There /s eorrespp/jc///?^/y j/va// ir^r/^/^on//? 
ffje rafe (^f //7crease ^f the i^ncTion. 7?je 
Thve (t>r Instantaheous) Rate of Jncreasb 
of Cf funct/'or?^ f^z)^ /s c/ef/nea^ as 

Th/s \A/ho/e series of oper at/on s /ir c/ent^ 
ecf f)y as/mp/er sy/rih/, ^yv/t, /f^=-fOc) 

£-fCx) or -^ 
an^ file resu/f /^ caM ^e De/?iv^77V€ of 

f{X)/Corofy), WITH FBSPECT TO X" 

7%e fo//^m)7y are e^c/Jm/e/?/' no/at/o/?s* 
. True /TaTe = tni^i^^oiAvG. Rate] 



13 

with r espect fox, M^fQ^XTT, '^^^ 
jdQ- X-f 4?c+/ *" «7T. /fcc/c/ce fo common ^enom- 



^tarnp/e: ^Jhcf /he Pi/e speed erf a body 
yfhic/? mores ^o /^af /f traverses id/sfa/ice 
of i^c^-rO^oiilmVcs /h ffje first 'X /?oi^rsi 
Sohtior?: Ryf s= %^^(\^ncy T^je/? ^^^t Av6 

iram/)^y ^ ^c(z±jc^±Axii±s^ 

Tahhp fhe /I'n?/'/' i7S ^e jhtertra/ A%^0 
True speeds gi=-^^^^i per hr, 

B-/7 /fQ^W> work oaf ^^^TTJT 

B-18 /f Zr=: ^^ ^/hd ^ /h terrrjs 
of X ond ||- /h fSrmQ of x 



14 

B-^0 // fhe o/eni/ty of a //(/i}/ //? a cyj/rr- 
c/r/ca/ jar ^ar/es rrom ffie dcffom t^p so 
t^af me mass /h /he Jmi^r 7(> a/rj. (>f /^e 
J^^ <^ ~i^ ^/27/7zo s'/^oyif that ^e 
c/ens/lfy j/i a /^i/er % cm, fr^m i¥j€ hcft 
T^n?^ is \^o-r ;| Oc-i-^yjo/ns. p, au cm.j ^sf.cf^ 
b^//7y we cross-s^cf/o^ of the Jar 

B-5/ //J= q^r, a/?cta::. ^fan^r^ ^-^t 
-f/ha/ the tPr/?7c//a ror ^/c/i. 

B-^^. /ff^e /?7ass-i^o/cme for/vi^/a js f1* 
^V^-/, i/vhat is the tornju/a tor Gtens/^P 

B-^3 /Fepresent the ^pacc-h/ne tormuja tor 
a fcf//fhg hodv^ S- ir^K^ ^ ^rap4 ^^ 
st70bv t/jat w s/ffpe of the ^/r^erit ctj^m 
to the cur/e. at any.jbo/rrf /s efaat 
to fhe ira/^e ^/^r" tor ttie \^toe 
of t c(;rresponc//ny to fhe po/pt fakeri 

15-^4 The spee^-t//77e formpta /s i^==-im 
what is fh€ acceteraho/7-t/h7€ tormo/a^, 

Br^^ tf S^7^^, ana/ h. ctepena/s o/j the 
t/Jo/epe/7o/e/7t irartat^te I (formuta^^ot 
y/\/e/7l shoyy that J& = (M/?-/; ^ . 



15 

Sbvb:hal Dependent Variables 

// seirera^/ i^an'a^/es c/epen^ (/por? a s/h^k 
/nc^€penc/e/?f i<ar/W/t/e we may eamp/^/v 
fhe/'r ffa/^s ^fc/?a/?^e ii^/?d ^efa c/er/mffye 
ofcir/7y (^ne yy//)h respect /o ffnc/ ofhc'r. M/he/i 
fJie mere me/7 f (?f me Indep, Yah. /a^hhes^ 
o// me correspO(?afjh^ /hcremenTs w/// 
\/a/^/sh, //e/7ce if a,ir;i^x,.,..are re/ated 

specifjed c^s ap/procroh/hy :zero. 
Th^re^ore ^ f^ese e?cpres3/'on s'- 

af^e e<7^/y^/^^t fo 7^e e/(pr(^ss/pr?s\ 
wh/oh,b/ DEFINITION {pa^e 10 )^ ^re 



Sftjce if /s r7cit necessarc/ fp jyoecify 
ir/7/c/7 /hcrerr^ent approaches 0, tfre syrrjm 

m/J jbe ^/ve^ th/s mc/er rr7ea/?f'y7^x 

TAKE THE LIMIT >APP/?0>^ CHE D WHEfi THE 
VAR\00% INCITEMENTS APPROACH ZERO 

Such c? //'j77/f of an^ of/f/ere/ice '^i/o- 
t/er?f /7?c/st be a aer/raAVe 



cp--' 



16 

Limits of Products And Quotients 

fherf €^0 a/7^ z'^O /h ^ass/n^ fo //m/ts, 
JT^e Proouct 14 v — OC+eX^ ^i) 

Henct X ui/-^ A& 

or Clu xi^]^ [£ u] ^ L jC u] 

Limit of a Product ^^^-rt^ooucr of Limits 

neQuoVENT^-4^ 

Hence [ LL ^ A cpfotfA^^nU 

or f T-^1 ^= L L t>^l ^forovieiec^ svch a, 

J-'LlT'J U JC ^J XquoHent eK/'sIs 

LlMlTOFAQt/0Tl£NT =^ QuOTIEMT OF LlMlTS 

These two Theorems fea^ To fhese /mpor- ' 
taf^t re/aT/'e>ns Setweer? c/er/vaT/y£'s: 



i7 

The DEffivATi/E '^ /'s £7/? abhrei^/at/on for 

fraction^ — 6t/t any >s/h^/e nf^'mber may 
be reyar^ec^ as a fracf/hn: t^^^<yr^ar^ 
Ther^ ar^P /rrony ff^ra/ilf^^es />? reyan^/nf 
^ as fhe <^i/a//e/if c^ffkvo numbers^, ^y 
an^ dx, i^/?/c/7 are cra/h/ biFnf^turiALS, 
/^ ^ is a/jy rar/ah/e connectei^ w'/fh yatK/^ 
(it /vay c/e/iate %/tse/f, ory,jh some case^ 

/n y^zf DvrFEREMTfyqt M{?TA7i0M fhese becojue 

where each Different/ al s/yn/fyes f^e 
Derivative w/fh respect p the samt 
V/^n'a/b/e (r^c^f/hd/cafs'a/j fh each case. 

The c/Z/ferenfla/ r?oTai?h/7 mahs //' nted/ess 
/d afAcr/r77/'/7^/e /'r? aafkiaf?ce. yvh/'ch /'s fo he 
P?€ INDBP. VAR' af^r if /^ choser? I^5S TO 

Derivatives by i//r/f/hf /h //s c//'f/eref7t/'- 
af c/ne/er eac/r d/'mrtrf/Tal appear-I/jy. 



DlFFEF?ENTML FoRMULfi^S 

To aefa -forrrju/a for fhe o//'ffereniTal of 

cf -furjct/on fof/o/v these Six Steps'. 

/. Suppose ihe Indbr\/ar. 9/^^6/1 o/i /ncrPrrient 

k.Fihd fhe ammnte^ \^a/ue afth^runcr/on 

5. S/btrffcttoferfhe/hcre/7!eritafm^r7cm 

4. biWe bi/ me increment of The Mep- i/^r. 

5. Passii^ ihe/ifvf't'as/hcremenrs ^/^>"^- 
6.Mo/tipfy ^ di'/ferent/'of or Mep. /ar: 

TTjus to f/hof fhe differential ofu^i/^ 
I. Call ^e Ind. Vgk t, arydfetif become t+A t 
^. T/;e^j andu^ become t^Ay and ir-^Ai/: 

6* dy =■ ^ o-cdi/- 

five eas}/y remembered /hr/r?c//efs enable 
one to cf/fferenf/iffe^ am (7ffe/:>ra/c ^x- 
press/h yy/fhoot ^/r/nf fne s/x sp?s saj^ 
or7ce for afl /h edec/ue/hf fhe for/Tn/f^^- 



19 

In thes^ c cfrjrfn represent ar?y CoN5T/^NTS 
fcrmcfm u c7/?£/ V- represent a/7c/ VARIABLES 
Formulas 

ir dCu-^ I/)— du \ dw 
SSL d(cu) ~ cdu 
ir d(wu) = wdu ^u di^ 
3z: dicT) r=. 7% uP-^' du 
Phoows 
r Lety^c, Then (steps\ and?) rio i-nat- 
ter fn^hcff /hdepenrtent ira r/ab/e mou 
charj^e^r/ d^es npt change af a//^ c^na 

ar?i^ (6) ^^, r/?at /s d(^ ^ O. 

(6) dy, P^ot /s d(ui-w)^dui-p/t^ 

^ let v^^cu. r/7enO>^,^) ^^="5"^"^ 
C4) AuM= CAU/Aocj Cs) UMtc^ c du/dr, 
C6) ^y, that ts dCcu')- c du 

m. let u^ u^u . 77?er?^ (steps I ^nd^) 






(6) d^, Ijhat /s d{^u £^)^ c^a^cc + ly ^(^ 

s: let^k u^, 7f»REEO\SBS. fyrst case is 
when 71 js a posJf/ve yv/ro/e /7umben 

Bmtn.theprem a/i^ cance/ (/) 

^'^ M-^^^^ ^""i^^^^^^o 



Seconof case /s y/hen n /s a^ fraC/lfinj^^* 
Let y^ u''^^^ _ ^ 

The/7 u^^ -^ . ("TP^at a^a oase 1) 

p -&-/ 



IJjirc/ case /s kvhe/i n /'s fte^af/Vc, ^^^nv, 

let y"^ u-'^ 

Tjhe/?y»u'^—f (Treaf/hefiroc/c/cfbc/ ir) 



rn 



Comparing ffje second gnd ffj/'r/^ cassis 
w/'th fhe first jt /s s-een fihcff n//f ether 
n is p/i^s Pr m'/7i/s^ /v/ro/e or frcicP'orja/, 

Formulas /or dd), e^(u-hc) ^ d(u>^ir^w) 
ot(j^), c/(Vu) are sometimes aMec/i bur 
fjhey are not /?eec/ed as r^ mac/ re^arzt 

u + c as u-nr and c/se a ar/t x 
uww as (uv^w anaf ase JX. Ti/v/ce 

/i^ as uir^ anct i/se rs: anct s: 

^ as VL^*- arrd i/je szr. 

Fractions m w^/ch e/tJier fiie /yumer- 
a tor or tfie ote'/TarTr/rra^r is constant 
shoe/lot h frea/^af fh^s \ re^ar^ 
^ou as -JciT ^W ase HL 
Vw as a 6/-'' and c/se uc cmd sl 



7.^ 
Der/V/Atives er OiFFERENTiAL Formulas 

^rerc/ a/pejbra/c expr^ssm is prJmar/ly e//het 
a SuH, a PROOUCT, or a Po^^h, of consfo/?Ts ^r 
^oricwks, a/id ma^ be hrt^c^^/it (//?afer^sm7€o/?c 
o^ me fyr/77i/Jas\ x'/bx. /y'rsf ryn/e aioi/y/j 
DjFFEi?£Nfi/»LS by mea/?s of fjhese fiftmi/Jai 
fhe/1 mss 'fb ifie re^^c/ireJ Der/vat/^^es by 
St/pp/y/^7p fhe same af/f/ere/i/^'a/asaafenom- 
fnafor fo eac/? af//fere/?/ra/. T/?e /a/iet- si^p 
may i?e re^art/ea^ gs oivioing iyi^e ^/'ffen 
e/pfia/ (^r ^e jnoepe/vdent v/IRMSLE. 

T7j^ c/er/Vaffve (;f a yc9r/ab/e yv/Yh fi^ 
spect to /fse/f /s (//7f% : for 

So if m cfifftrenf/al af'th^ lHt>EPmt>EHr 
Var^ occi/rs /h cf/7 e<^i/ai?i/7 // g/zW^^^s 
Ot^t yiz/je/? w^ pass io c/eriya/j'ye^ (7/7^ 
feak^es the factor I /h its p/ace. 

^xampie\ y-C^'^-^y^, f/hc/ ^ 
by r au-^ZCx-f-lp ^(^t?*-/) 

ierirat/y^s *^ 




pass to c/e^fn^^s. ^^ ^ 

In worhj7^ oc/fsych e^a/J7p/es^ //?e /earner 
/nusf be co/fscioi/s at ei/ery s/e/p of the 
parttcu/ar formula h Is us//7o. //"/>/7^/ 

r?ecessa(y, foi/ye^er, erce/pt ar f/rst ^^ 
/nc//cate ^ sfe/ps so fully as «7^^/^^/ 




/)/7J?r c/ff/lpre/jf/at/ffg ^ Propuct^ 75^(f j-z/^t- 
plejf yyay to make fhe a/gebra/c realc/c- 
T/ons is to Factor oc^f fhe neivly frjtro- 
duced poyyer of each fbcfpr, as cfdove. 



24 

Driil on Differentiation 

In ^e exnmp/eshe/oyy^ a,p,c, n^represent 
CoNSTAHTS. All fhe piiier [effers rspftsent 
V/^RiA6LES. Bif c//fferenfla/l(y? ^ mea/js 
af 3;ir, m^ii.v; i^enfif eac/j answer 9/ ym. 

C-( ^= ^x^-00-% ^ ^6^6%-<]r() ^^ 

C-3 2^= a-x?" +zb« + c , ^= ^ C«X + b; ^ 

C-4 ^^r^y^-Oif^'+i/). 2|=29(-4Hf3+>J -I 

C-8 .^ = Ca«? + br^*; ^- = f 

an^ dr. also die /n term of ^ ontd d<j. 



Z5 



0-18 o(^4za^^ff=b, ^.^ 2L±^ 



^6 

Successive Differenhation 

/I c^enVt^f/ve is itself a fi/rJcPoh that 
mat^ he ct/fffrer/l^ht^d /v/th respect to 
0/7^ re/at^d p^ar/'ab/e. /i// Gi//ttre/7t/a/smst 
he c/far?fei^ to the DEHiVArive noTat/cf? 
before fhe(^ are ^emse//es ottfferent/aTd, 

The $i/mho/ K%(3^) /^ i^ntten rnore bn'eflu 
In sinrilar my M^^- Jx^^ ^^. 

Examf^/e-, &iven, as abo^e, ^ 



C-30 f^^(^%^ ^^--f^ 



Z8 

SUBSTlTUTfON INTO DERIVATlveS 

Calculus deah with fhe rehhom b^tweety 
vanables as theu change, crnd not with 
specrQ( irqfoies of the yart'aiD/es, Accord- 
in^ly if o cQiculus problem <^ ives or 
rec^uilres special k^a/ues, GsNEffALFoffMviAS 
must be ^ot first'f Qnd the speeioi values 
substituted tot the yan'ables qs ttie last step. 

Bxample: The bead, B, mores up the u-axh 
^ at the rate et Sin, per min> aind 

oB dro^s the bead. A, ctlon^ the x-etm 

"^g-., A ^ b(/ ynearis of a cord 5in. tonp 
How fast is A movirtcf when 3 f's 
4 ihches tforn the ori^in'^ 
Solution: The coordinates oF A and B ore 
(%,,o) Qnd (o,q) and the general relQ- 
tibn betvi/een the variables^ iLand tj, is: 

Ttie dih.perrnin. IS q special v'oilue of 
B*s speed, and the general speed sdi^rdLi 
if t is the time e/cipsed in minutes, 
A'*s required speed is dx-rdl wheni^=4. 
Ditferentiatirq: ^Xdx -t 2udu ^ o .. 

Whencf-4, ^t^-^^^'^j on Down 4in.per mm. 



A m ahbreyCaHon ^r&^uenNt/ used for the de- 
nkQti\/e of o function^ f( ), )s fV i^in 
which the PntMB rndthctZ-cs tJiaf- the. 
f-ufictioi^ has beetr d /? Fe re n hated with 
respect to its Qr^ument In short 
f\-%) s|^ f(%.}. 



This notation is extended to hi'^her derlrtthyesi 
rc:^) s ^ f C%) etc. 

When the argument in a derii^af/i/e is 
to be rB placed (seii^ bif a constant) this notation 

f'cc) 
me^nsU^^ get f\-x) by li/orkin^ ^^^ ^H^) 
U'^^Sf^bsh 'tute c for the a r^ uwent^ %-. 

ExQmple: Giyen f(cj)= 3c^^- y*! find F" (^^ 
Soluh'on: r^^)£^ ^C^)- '^^^-^Jf 

C-3h f(X)=^') f'(^)-? ^ns.-^^*^ 



CS-j <p (X) ^ Z-x^-avc-f-^j 5how that q> 'O) = (p"(-k) '<^'{o) 

C'3& <p(x)=-Zx,^-S'k^-* 4j 3h ow that cp (i)^ 
(p(o) ^cp'fo) -^-^cp'^io) +f ^'''Col-^-h cp'Vo) 



30 



Shapes of Cuirves 



P/oth/i^ o function, fi-ic), b{j points is t^erjj 
mucPi shortened bi^ usfnQ fi%^ oind -f'bb) 
to cjet the Slope ana the 3hari^- C ar 
Curycff-ure ) ah a few points. Given w = fC>^ 
Aybvage Slope = rise -rrun-^u-rA-X' 
.■.Slops, 3 ^ X4fe = 4i-= fW 
Npte thah inhere the curvoihure is 



DOtAE-SHAPE 




BOWL-SHAP^., 



>$ is decreasing and 
^<g = i"(%) is N£G, 





Sis increcfsinof and 
^^^V'cx) IS Pos, 



It f'O^ comes out -zero, ta/rt points q 
htth to each side. There are Four cases'. 




Tht ti/^o middle cases Qre called InflbctwnS. 

CKdmpJe: Plot u=*^-^ from a f^w points, 

Solution: Put ^-f^f(x) 

FIRST step: Ploh points that can he cal- 

culole'd bif inspection: 

in this CQS&, the%e\ "t 



31 

SfcoAfD Step: Get genera/ formula for slope 




)^ 



Third Step: Get f'(x) and plot stKfpes. 



^^ 



> 



Dome; In f lee; Bowl, ^ 

FouFfTH Step: Inspect the formulas for gen- 
eral facts cibout the 6i^t}3 of fOffOj-fo 
in this case f() \s i: Qccording os % /s 
±, cfnd ^/ves bow/'sticipe 
to the right and dome- 
shape to fhe left. 





Plot the graphs of fhese e^^a - 
fionSj calcu/al ing a few points onlL/l 



C-4/ 



3^ % 



C-43 ^ = 7"^=^^ 



3£ 

Ekact Rates or Increase 

ftie rate at which any quantity Q. [whicti 
moiy represent a distancej a volume, o 
Speedy etc. J increases as compared 
W/M anot'h^r ranabh, v, [wtiich mat/ 
represent Q hpse of time, a changing 
radius J e^c.j or ont/ cac/s-e or concomi- 
t<mt of tt^e change m d) /s rep re- 
J^i^ted by I ^^ Derivative 

for tfi^s ' IS bcfiNto cfs ttie result of find- 
ing the ratio of the increase in Q to the 
Increo^-e in \ and then finding the volue 
toward which this (hveRAGE Rat^) tendsas 
the mt&rvQl For vYkioh it is CQlcula/ed is 
Ivade s matter cfn:d approaches zero as 
Q hmit. As on pag^sio^tt, so in general: 
AS- = Averagetfat^ of increase of Ql 

\ in units at £L 
-7-^ = Irue ffate J per unit of /, 



In studying calcuius, hrin^ fo mind 
continually various in7portant «nd 
familiar physical rate -i^uanfi hies, of 
the same nature as derivati res. 



33 



These are well known definitions: 



Av^, Speed - 
A^g. Accel, - 
A^, Slope -- 
Ayg, Curt®M -- 
A\^, benslfy- 
Avp. Fiyiver = 
Ay^.Crc?33 Seer- 
Ay^. Pressure=^ 



■ Length of motion 
-Gain in speed 
■Rise 

'Amt. of flow 
-Mass of plec e 
-Work done 
■■ Vol. of piece 
Force on Area 



-r Ti'm e of mo. 
Lapse of time 
tfun 

Time of flow 
Vol. of piece 
Time tali en 
it& Length 
the Area 



Ay ^, Rate, ^r \~\ Increment ofl-rflncr of 

Oil Quo, J lDeP£ND.\/AR. ) {liyOER )t 



an 

Var. 



All such definitions lead to exact for- 
mutos ojiyin^ Bxaqt Rmes qs l^ERtVATivEs. 



em. per sec 



= d 



£ 



di 



Slope ~^~W^~IE 
cm. per cm. 



cm. 



i = time 
sec. 



u s rise 
cm. 



X 2 run 
cm. 



hcceleratton ^ cx - -^ 
cm . p. sec, p. sec. 



Cross-5ecl/on^ 3- ^ 

3q,cm. 



Y s speed 
cm. p. sec. 



I zl/me 
dec. 



V^ Volume 
Cu. cm. 



yC'S length 
cm. 



Pressure 
dtfnes p. sq. cm. 






F = Force 
dynes 



A = A rea 
S(^. cm. 



34 

Derivative Rate Problems 

Attach an thes^ problems in this gen- 
eral waj/ : 1^ , note what vanahles are 
invoiired; Z ^, e/ press tlie reta- 
hons between tt^em by means of ecjiua- 
tions; J^- note what rates are in-, 
volved) ^-, obtain the genera t tornm- 
las for such rates; and fast, subs tl - 
tiLte given values of The Variables. 

D-/ /4 pebhJe drop peel into still water 
creates a circular disturbance whose 
radiad iengttiens iZcm.p sec. At what 
rate is the distur bed orea increas- 
ing when, the radius is one meter^ 
Ans. About yS sf meters p, sec, 

D~<3 The diagonals of a recfang ufar 
plate are increasing at the rate of /5 
in. per sec. , and the ptate is lengthening 
at the rate of Zi in. per sec. When its 
d intensions are 6irK)^tZin, how fast is 
the ptafe narrovi^ing or ividentngf 
Ahs. Narrowing '^ in. per See, 

0-3 If PN id normal to the curve 



u* = /2 ex at the point 
ordinate is d , a n d 
F>M is normal to the. 
X-ayis, colcula he tt)e 
distance f^N. Ans, 



35 

f^ whose :x,~co^ 
P 




MN^ C, 



0"^ A body starts from rest Qnd in 
t seconds cf a ins Q speed of 

f ( ^ -^ QitTT i- ^V6)ft p, sec. 

Ptot the time -occe I era tion curve, 
for the fir^t five seconds. 

05 A vessel in the form of a n inverte d, 
circular cone of semi-vert'ical 
art^le '4^^ IS bein^ fitted w'lth 
wcffer qt tine uniform ra te 
of 3 ca.cm.per sec. At what 
the surf^ace rising when the water has 
reached the depth of x cm.? 5cm.? lOcm^l 
Ans. *&3^X^; .033-^,0085 cm.p. sec. 

D'b A mar?,bff hi^h, wot/(ing ai the 
rate of 3i m'l.phr., passes under a. 
li^ht Id ft above his ^-'"^ * , 

pat-h, which IS shra'ighf .-'"'11__Jl. 
and l^veL Get general " 
Formula in terms of i^ the number of 




minutes since he was under the jtcjht 
for the length of his shadow ond 
the rate oit which it- is Ichythen/n^, 

\^-J In the same case as D-^ fifid 
the speed with which the end of hi>s 
shadow mo yes aton^ the ground, 

D^Q A goisoline honh of irregular shape 
is heiing filled. The n lumber of goJtons 
in it when fhe ^QS^h'ne stands a/- o 
depth of xft is i(5^-fO?+1ik), Ihere 
are about ft gallons in a cu. f t Find 
the area of the loottom of the tank and 
of cross- sections at depths lft,Z1t,'^tt, 
Ans. .Z3y ,30^ a.,36, ^.?3 5^./"/: 

D-y The cutres y^ § X^ ^ x^ and ^ = 
^%.''-^X-tZ, intersect where /C^Z. WtJtch 
curye is steeper at fh/s point? 
Ans, The/r slopes are IZ and 1t%. 

D-fO The side of an B<j[uilot, Trlan^U 
increases of a rote of /Off. per min,, 
the area at a rofe of I0s^> //. per 
^ec> tiow large is the friang/e? 
Ans. One side « 6f>3 ft, 



5r 

D-ll The e.quations for y' % 

a proj ech'le ore Z^'^OOi 

and ^ ^ 3^01 -IH% tn feet -f 

QHd sec. Ff/7cf its d'lrechion of mo hi on 
wheyi i= 5 sec^^ lOsec^ IS sec. 
Af7s. f24'' up, S" up, If dou/n from level. 

O-jZ The cosf of digging a pit^is'^ip 
mulfipfieci hjf fhe cross-sec. //? Sf, yds. 
times ffie sum of fhe depth in (/c/<,^ pic/s 
one-tenih its square. /// w hut rote a/oei 
one pQ(^ f^or ej^cavofion at ihe bettom of 
a 40 ff. pit of uniform cross-sectionj 
/\/?s. lot cents, per C u. ft, 

0-13 The point P moires o/on^ the line 
u-6 at the rote 

of Y ^^ii^ P^i^ '^^^• 
Hoiv fast is the 



-.o—> 
T 



.o; 



M 



middle point of OP receding from the 
point {Ofi) q/ tht instant when OP 
=^ 10 units'^ Ans. ^J units per sec. 

D-z-f Which increases more rapidlj^ 
a^ -^ passes through fhe i^ QJue %^'^l 

(Z^)^'^ or ' A-p^-^c. ? 

A/?s. Their rates ctre as / .' t»44. 



36 

D-/5 A ship sa/Js due north 10 tn. perhr. 
A steamer^ j mi, south, 4^ mi, \ 

west, steams due east at ,-' i 

twice that speed. At what o-<-' i 

raie 13 the dist, between them decreas- 
ing? How far does the s tea met ^o be- 
fore I' heir dist begins to increase'^, 
Ans. t6./ mf, per hr; 30.Q miles. 

D- 1^ A wa/e crest approaches a li^ht 
house at a rate at 4&ft, per sec. The 
If^tjt is 3 if ft. above the sea.ttow far trom 
the foot at the tower is the wai^e crest 
when it 'i6 qpproachinj^ ihe li^ht 
Qt a rate ot 44 ft. p. &ec? QZ.h H 

t)-}J The time- distance formula for a 
moving point is % ^ Zi (l-z)^ Work out 
the fime-speed, and time- ace eleration 
eauations and plot all three ^^^0 to 1-i. 

D-18 Ihe total force, on the upper lA 
ft of a recta nq. dam 30ft lon^ is 
<fOO ^^ Ib^. What IS the pressure 3 tr, 
from fop of the dam ? ^30 lbs. p. sf. ft, 

b- t<f A man, iff, fall wallas directljf 



39 

Qwau trom a lamp- post, 10 tt. high, at the 
rQte of 4 m. per hr How fQst ,,i) \ 

IS the mtddle of h'lS shadow ,,^''^[ 'o^^*' 
moving along the pavement ? ""' 
Aris. J mi. pet hr. 

D-BO If Q,^ dx^a^S) and S^fh^\ Und 
the rate at which Q. increases when 
X is i and iS mereas/t?^ at the rate 
of i units per $ec. 



J7 ' ^ ' ^ 

Ans. »hoe units p.sec. 



D-^f ^-s amaj? waths out ahn^ « spring- 
board, one end sinks to Q distance of 
u^-i-g^(x-^Z) inches when ^^ JJ, Z re&t 

from the wharf end, _^pI:^I^-^..-Q 
It he moves aiongi lllljllh ^^^ 

at the rate of t ft "^ 
per sec^ tjoiv fast fs the end sinking 
when he starts? When he has^^nefOft? 
Ahs. ond d.JJ ft per sec. 

b-ZZ The equation of the c^urire SPQ.i^ 
X% ^ i^X ^^, Thetan^^nt 
at P who^e ^-coordinate. 
IS (t, IS the line 7? T, 
F^nd the length of R T. 




40 
D-2 3 When a ball is thrown straight up_, 

it repiches q height of (h- lfl^+ iA-Ol) 

feef It? ^ sec. When does it's Speed change 

from upt-(? down? Ans. i= ^. 3<3, 



b-s4 Atocf in the form of a cylinder, IZ 
ft ton^, ^ft. in diameter , consists of if ear- 
rings whose densitif increases toward 
the center so thai- ^ 
the mass of a core ^* 
whose radius is x in, ^ 
is hdO x''^ (20i-x)] tbs. Find t/?e den- 
si tjf Gft the center and at the ibor/r. 

Ans. h3 Qr?d 3Z lbs. per cu, ft; 





Hote: In some problems data are ^iv'en 
which would he difficult or im possihie to 
ebtoifi in an otctaal cose, while the (^uan- 
titles asHQd for could easiij/ be found hj^ 
direct measurement, it is nevertheless 
worth wh)le to sol ire such probtems for 
the sat^e of the tig tit thecf throw on 
the reversed problem. 



D-ZS PT/5 ont/ ta n^eht 
to the cur re ^* = ^ a^. 
Prove that Om = OT. 




D-^^ The speec/ of a meteor before it 
reaches the denser jpart of the earth's 
atmosphere is given //? ft* per s^c- <fs 

z o or 
where r is its distance from the cen- 
ter of the earth tn feef> F in d //s 
acceleration Qt a height of ]5coO mi, 
above the surface of the eorth. 
Ans. Ji ft. per see. yoeraec, 

D"<27 If ^ i^ ^ number oi fL^ t, o/ 
number of inches, m, a number of mm,, 
and' s a number of sec,^ write down equa- 
tions of relation and from them deduce 

elm "^ <ls 

D-26 An automatic record dhows that 
ftie worl^ done i)y a certain engine in 

% hours, bejinnm^ af f^^^'j% ih. 
\Q%i \0 ^ 5h -%K )\0 nibs. 

Find the poujer bein^ lAsed at <f Attend 
at /0.30AM in horse- pow e r -s , one horse^ 
po^er being eaua/ to 550 ft lbs. per sec. 
Ans. 1^4 Wcind ^05 tP. 

' D~Z<] FromRe^naulf's experiments it 



^2 

mppeored that Me mum her of hecit uhits,^^ 
tec^uired to raise I'he temperature oF h^ms, 
ot water from 0" to T^C^, is given hj/ the e^ua, 

// heat is bein^ supplied t-o BO^ran^sofwater 
at the rate of to heat units per sec.^tind 
the rate ot whi'cli tht temperature rises, 
when T= 5<9*' An%. ^^cfj^ per sec. 

D-30 A rectan^u/or plate is eApand//?^ 
at the rate of tds^-ir?. per sec. in such a 
wat/ that its dia^onats re ma in the ^a me 
len^th^ At what rate are the two sides 
chan^t^n^ when the dimensions of tine 
plate are ^ in. )i %0 in^'? 

Ans. Incr. *5ZZ, and Oecr. .10^ in. per see. 

0^31 The paint "P is on the curve u*= 
x\ at the point where ?t'-f A tangent 
js drah'n from V to the x-axis. tyhdits 
length, Ans . S.-fSZa unit%. 

D-3Z Assume ihat the work done in 
compressing a va por fr om volume K to 
volume l( is l5^[v^Tv, inch /bs. 3^cu. 
in, of vapor are introduced into an 
empty ct^linder, c^ so. inches cross- sec- 



4d 

t/o/7, and bif means of a pisfon this 
vapor IS Compressed to leu. in. Find the, 
force tiStn^ used at the end of compression. 

Ans, -405. lbs. 

D-3 3 The altitude of the sun is 30' 
when a bait is thrown verticaliy up to a 
hei^tit of h'tfeet. /tow fast h its sliadohr 
traire///r?j olon^ the ground j ust behre, 
the ball strikes the earth? 

Am 3. III. ft. per sec, 

D>3-f A circular metal plate /s ex- 
panding so that its rcrdius increases li 
m.m, per mi n. At what rate does it^area 
jncreose when its diameter is 10 cm? 
/4ns. '^.'jr s<f_. cm. per fn'm , 

D-3^5 * Vsin^ the term ''length of shadai^" 
to denote the distance between oi ball 
and its shadow on the ground, find a 
formula for the rate at which the shadow 
lengthens when a ball is thrown up so 
thai its he/jht is (dot - IH*") ih at the 
end of i sec,, the shadow mahinjon 
anal 6 of AB" with the around, 

Ans, if-liofs[^-^<lf.p,s. 



ok-er a ^ridg^e 



44 

D-36 A man /s walking 
at the rote of 4 m,p. /?. A 
bOQt passes under the 
bridge just below htm. It Xi 
IS iowecf a m,pM> anclthe 
CQnol is 10 it, below the roadwaif. How 
fast are the man and the boat separ- 
ating 3 minutes later? 

Ans, i mile per hour. 




D-3y A soap bubble remains spherrc^f 
and /ts diameter increases at the 
rate of Zcm,persec, At what rote is 
its voJume increosi n^ at the instant 
it becomes 15 cu.cm] 

Ans. £7.^ cu cm. per sec. 



igjram: 




D -38 Talcing data from 
ttie equation of the 
curve bein^ 

CQlculate the length 

of PMj which is drah^n perpendfcul or 

to the tangent at V, where ic^^, ,hj3 

D~3^ In D-3© calculote the distance 
fram M to the origin, 1.1^5 



45 

D-40 The cosf' per mile of- running q 
.5teain-^oah varies as the cube of tht 
Stpeed, Shokv that the cost pet hour 
ycirte% os the Fourth po^er of the spe^. 

D"^i A balloon rises to a height oF 
[lOm. -7^4^00 +m* 3 Aw//es in m min. At 
what rate is ih rising at the end at 
the first half- hoar? An^, About Jm.ph, 

D-4t Water is poured ih to a conical 

cup at the rate of f4 cu,cm. 

per sec, and fills the cup in 

11 sec. if Its depth then Is 

Jem., how fast is the water rising just be- 

tore ft overfloivs? Ans, ,Z/Z cm./D.sec. 

• 

D'-fS At what unites does the cune 
u-B-x^^-^x cut the -x^-axls? 
Ans. -f- QS^'tS' and - 30" 35' 

D-44. A rod glides through riti^s,atA 
and B) which slide on fhe axes, 
and at C which slides on the line 
xti'O, If AO-[^-^l^)cm.,OB'ylClm.- 
ffnd speed of C when'i- Z seconds, 
Ans, <fJ2' units per sec. down. 





4^ 

D -^5, H /s a nurrfjber (pf /7oc/n-^^ /t$ ya/c/e 
//7 m/}?(//es'j S, //^ ira/ue /p seconds', set 
c/t?i^/} the ^^i^^y/o/js ^j?/ re/a/7or7 /?/7/f 
fi^o/v th^/7? a/€£^c/ce f/?^f ,^ 

D "46 4 /ve^^^r is fa///h^ ^3 ffie e/^rth. 
Its c^/stcf/ice -fro /7? the ce/ iter of the 
earth is [7000 - ^icoooV-V^Jm/t^s. ftie t 
beJ^f ,^ P(//7?her of seconds. Smw that 
it s/r/h'^s tt?e earth yvhs'/? f~ looo. 
fv/fj? a speec/ ^/ 80 m'/tes per p7/mte. 

D-'^f Locate ttie /?e/hts on the cc/ri/e 



ar ir/h/ch the 7a/7^er?ts / — 

Gre f?onzonta/. Aps.(i-\4^(>)s(^f^^4' 

^-^& 7?;e pen^eTer, p^ of (7r7 I'soscejes 
trian^/e re/77a//?s co/7star7t' /?X 
wh!/t the a/t/f[/i^e /h- x J^x 
creases cff a^ rote of S cm. per sec. ^ M 
what rate /s the c^r^ea wareas/h^^ 
A as. (^p*-- lSiK^)s-r4P sf.om.per sec. 

\)-^^ A yve/^ht haafis fr^m a spr/h^ 
an^ r/ses a/r^f fa/h so that as speed 




47 

/ J Z Vey-i/* cy77. per sec. , ly ^/77. being 
its d/'sronce fron? the upper ^B>\% 
end of the sprinq. i/^ork ouf ^^\ 
a formic to for the acce/eroTion, «<, in 
terms of y. Ans. a- ^(^-^)om'p.s. p,s^ 

D~50 In the ease described in D-4/ 
find the rote af ivh/eh the 
v/s/bfe surface of the earfh 
/^ increos/ng at the f/me \^'^ 
menthned. /^rea = RnT^x °v • , 

f\/7S, -^6.3 scj/irT?/, per Sfc. ' ^"^ 

0~5I At what rofe does the i/ofume 
of a cone increase yvher/ the aif/tude 
o/rd ttie rod/ us of ft?e base /^f\ 
increase at the same ra^, /^LT^ 
S cm. per sec, eocf? f 

Ans: ^SnC^rh •«-r*j cu cm. p. sec. 

D~5^ ri?e atimens/ons of on ex pa/?d//7Q 
C(///nder are r^{\^+*^^'^^^)crrr. ^^___J^ 

bein^ t* C(j. temperature, if^hat ^ — '■ — t 
is the increase in i^oiu/ne per decree 
gt o femperofiyre <=>f 50* Cg, ? 

Ans. /^8. eu, cm.perdey. 






.,M 



AS 

D'SS ^// is forced c/nofer a p/s/bn /h o 
cyl/hder dt a rcfte inverselu ^ff]^ 
proporf/ona/ to the he/qht, h, 
of The pisfcn. If the piston 
is r/'s/nq of tine rote cf r ^/77. _ 
per sec.y y/herr h-Scm, f/'nd "~ 
the rate ot i^h/ch c/7 is co/7?/n(^ /h 
when K-\0 c/77. The pistons cross- sect/or? 
Js '^sy.c/Tj. /^r7s. ^ cu. c/??. p^r s^c. 

D-5^ A tr?osf /s be/'n^ hffed hi/ c^ po/r 
Of f/peprr: /1 8 = BC^Z5 ft f/oyy /^ 
fast is M being raiseet i/^hen / h\ 

f\ and C ore ^0 ft. apart a/?d K ^ 

be/nq drawn toqetfier by a fctcA/e 
at the rate of b inches per sec/' 
Ans. f,5l inches per sec. 

D~55 The area cf an e//ipse i^^hos-^ 
dforrfe/ers ar($ p and a in. /^ j ^^^N^ 
/s 4 ^P<^ </^in. /^/ yvhaf vT'T'^^ 
raT(? itf its area incre(7s~ 
iny if, t beings a time in seconds, p 
^ ZX*^ and q=^3f-i f 

Ajs: rrt (qf*- f ) Sf. in, per sec. 

D** 5fc A white-hof bi/ttet /s otrcpped from 



the ceUipc^ of a dork room $ sec, o/fbr 
a cold bu/fet /rr?^ o/rqopea/ from onothef 
point on the ceitinq lo ft. f^-^J*" 
away* Tt?e fatter costs- a 
sfjactow or? ffie woff 5 ft 
away. Tofte the cffs^fance 
faffen as ibft^ fhe si^i/ore' 
of tf?e member of s^^conats offoft/n'^, 
Ffncf t/?e posit/or? y spe&cf, occ eterof/of?, of 
tf?e sf7Cfcfokv wf?en ttie hot f^ocf^ hos /often 
for or?e seconof, Ans. ZZ*^t ft betoyv 
the ceitini^; speed, 58 ft per sec. c/oiwn; 
occeterof/on^ 3R ft /oer secper sec. cfown. 

D- 5 7 S'how that ff speecf /s prvportfon- 
of fo [distance]^ , occeteroffor? mi/sf 
be proporfrbncff to [speedL'j^ 

D^56 4/7 irre^i//c?r r^^s^k-o/r /s ^e//?^ 
fi/fecf. The yofi/zne of tvafer let //7, V 
Oi^. ft» rc7/ses the hef^hf of the /\rafer^ h 
ft accorat/n£^ fo /'he fbr/nt^/a 

/vhcf the c^reot of the pvc7/^r' S'c/rface 
when sf^ r?7////o/7 ct^h/c feef of wafer 

hare heen feh f^- . .,.. ^ /> 
4/7.S. Z^/earfy Ztmyf/zcn ^f- ^^' 



50 

D-5^ The rope fiSC rof/7S o^er o po//ey 
of B Gncf /s fasfer?ec/ to fhe 8 ?)* 
boom of C, The rope is wo(//7pf 
/h of A , 3 i ff. per sec. ^okv ^ 
fosf is the b/oc/fi L, rising 
ly^en BC is /eyef r' Tife of/metfsions of 
the c/errich are: AB'^^^^f, AC^ZS ff 
Ans, i^.0^6 /t per sec. 

D-60 A tro/'n is l^ot" 5 i^] mt/es from 
the S'tarfihp po//7t of the end of t hn 
Get its speed ond occe/en in^nm of J 

{)''h\ ri/7d the tirr/e bety/eer? stops of 
the t^air? descriheot in O^bO. 

A/7S. /? hours, 

D^b^ /%/7 c/oes the D-feO froirr 
ftrst t)e^i/? to slow Joy\^r? of^r s-tartin^P 
Ans \ hr ?.J/77//7, ott^r sti^rhhf. 

D~63 An etectrot(/h'c oett deposits ^^iC^cu 
cn7,ps. 0/7 3C/77, of irirs -^m m t^f'ctr. ttow tost 
is the d/ofr7der i/pcreas/r?^ whe/7 the yy/re has 
become I mm, th/ck? 

Ans. ^^ \cr^ ofj7. ^Gr- soc. 




51 * 
f^AxiMA AND Minima 

When a foncf/'o/? thot has been /hcreos- 
in^ begins to c/ecrease, /^e \ra/pe at 
wh/ch it turns is ca/lec/ c? tAM\fAV\A 
~ vifiefher or rrcf there mot/ 
be sf/// former t^a/t/es of 
tiie ft/ncf/orf. The ierm 

n?oxfmi/rr7 is L/sed in a 
refative sense, the yaic/e 
^c co/ied bein^ ^recf/er fhar? t/7e 

Voiu&S or? how sides of if, t^herr 
the fi/nction /'s re/ore sfLnfeoi (^rcf/Qh/cai- 
/(/ ar b(/ a taSk, S/mi/orit/ %.) iscaiied 
a Minimum yyi?en ?c-ol /f f(a) /s iess 
thQf7 ttfe vaiues of f(x) yvhe/? % is a 
/ittie less or ^rsffter tho^ a. 

in eie men tart/ wori( ia^s dea/ h^/th con- 
tin t/oc/s tunc tons which hoi^e cont/hu- 
ous derivoti\/es. The ora/oh of st/oh a 
fc/nct'on is o con/?h(/Ci/s iine w'lthouf' 
sharp it/rns Hire a contoor iine /h o 
ro/iing coun/r(/. Or? such c? i/ne the. 
points- i^hose- aiiitudes (yo/ues at the 
fyncfion in the curve u= fiX)) ^f^^ ^<^^' 
imo or minima ^ wiii be at the tops 




5^ 

and ijotfoms of the unc/u/off/ons. At 
these points the curiae ivitt be Level ^ 
that is the stope of the tonosnt, f V )^ 
yviit rr o . {But the stope i^ii/ at so be 
r=iO Qt a Terrace.) 
if then we ha\/e a 
function, ■f(x)j on(3/ f//7c/ 
Att the ^ po frits at 
l^v/7/ch -f'i ) = 0^ thts ifst y\^itl fnctuc/e 
all points at lA/hich fc ) is either 
maxim i^p or mfn/mum. These points- 
ore cafteot Critical Points, fhey may 
be sorted bi^ i^hichei/er of the fot/oi^ 
in^ t/jree methoets is mosf conyenie/7t 

Tms^\\^ETHOo Of Testing Critical Ar&umbhts 
Catci/tate the i/otoe of fC) of the. 
critical point onct compc?re with yatoe§ 
f^r Ne/gh BORING /Arguments^ less tho/7 and 
greater thc/r? the critical argument What' 
' is cotteof the Neighborhood 
of a cr. org. is that ranye 
of arguments mt separated 
from it bt/ anatf?er cr ar^, Ir? ctetenninin^ 
whether f(d) is H/n, c ord e are neighbor 
tng ar^t/ments: compering f(d) with /"(a) 
would lead to no conetus/0/7. 




55 

Second htrnoo or Testing CRmcAi Arguments 
rfnd the slope of the pra/^h C^ote of in- 
crease of the fi/ncTi'on) for Ne i o H Bo r ii^ 6 
orgumenfs'f chosen (as In the first 
methoaf) (greater om/ /ess than each 
criticai \/aiue , one betv^een ^ach pair 
of them. On/y the s/yns of f() 
meed be deferminecf td sha^ kvhefher 
the cc^r/e ny/7S up-/ei/-ef~cton'tr (/nax/- 
mi^n?)^ do^r7'/ei^e/-up(r77i/7//77urr?l ^r 
L/p~/^i^e/-up or c/a/y/7-/eire/-^o/yn^ as 
of o fierrace pa/'/yt 

TwHX) Wthoo of Test mo, CffmcAL Af^ouMmTS 
S^hsf/'fi/i^ the er/f/ca/ ar^c^m^/ri^s /n/b 
the seconaf cfsr/yaffi^ey f V )^ anat if 
the resu/i^ fs Plus /ve /hare a ya//ey- 
bpfih/??,he/7ce o minimum', if the result 
is Minus pre hoye a /j/i/-/dp and f^ence 
o max/mofm: ^^ and t^, tr -f() comes 
out zero yy/yan a cr/t/eat arpc/merrt' 
fs Si/bstitc//eof» fry NEIGHBORING va/i^es. 

&raph/aa/ repr^sent&fib^ /s of f/7ef/'7fat 
est as<s /n ^^^jT^^^j^ ^^^?6/t /77(2x/n7(^ and 
m/ninya and s'ftaa/d r7et^er f>e ne^/ect 
ed. ffey/eyy papes 30 and 31. 



54 

Example', fx am/he \^ Zod (:)^-'\) formo)(r 
So/uf/or?: 

The crif/cof or^u/Tjeafs are %-Oy-^^ 

As this /'s a/7 ///(/sfraf/'ye d'xamp/e 
eao/h .n76f/fOd/ of S'(:>rf7r?$^ ouf /^fje 
mayi/may eta, ky/Vf be ap/o//'ec/. 

1^ ME) HOD. {i ~»1S. Eas(/ va/i/es ^/^c 
in f/7c /htcri^o/s Sefwee/y er/f/ea/ n?/- 
o>es arpo' beyonc/ ore -1, -^^ +-^ ^ + 1. 
Co/co/o/e o Ta/b/e, pM/)h^ resi^/Zs r(Pc/^/7/y: 



^n/ermed 



-i 



-y%. 



+/^ 



+1 



TIT 



ic ) 



A 



/.5*6 



/.fg 



/. 



A^ 



\. 



Verc/^ct: 



M/^X. 



)\^htt 



MIN. 



This fab/e 
^ives fh/s 
plot: 



of /^he 



■<y 




55 

K"'*Method. Tobu/a/e f( ) /»/- crlfico/ ona^ 





' Iffy ^ 




i^f f*^ 





-w Ufff 




i/U^i^U, 


hJtrmec/- 


-1 




-Vt. 




+ ii- 




H 


f'( ) 


-f 


O 


- 


o 


— 


o 


-f- 


Slope 


^ 


— >- 


Ni 


— >- 


""^ 


— *" 


/^ 


l/erc/icf 


M/4X. 




Neittier 




MiN. 




U ) 


1.56 








.44 





3^ Method. Qdfa/'r? the seconc/ t/er/V(7f'/Vip 



Cr/f.Ar^. I -x/^^ 



4V^ 



rc ) 



--, ;. rN 



Q^'>y^t 



•+",..'. ^ 



V^raf/cf 



M/^X. 



-hO-,/.^ 



MIN, 



(.56 



f c ; 



44 



ne ■Pac'f thot ^i//s (7/7 c/ ^a//<$ys /77c/sS't 
a/fer/yafe ^ac/c/ shoyv' /hd?f x=^o /s a 
/^rn/ce po/nf s/hce //?e cr/t/ca/ po/h/s 
£^c//acer7i' ore ^^ M/ o/?c/ a ya^/Zey. 

/t? c/o//7y //p€ fihsf c/oz(Sn or so o/^ 
proi^/(f/vs, procf/ce /he i^se of <?// ///r^e 
/nefhf^c/s <:>f s^rf//7f cr/T/ca/ ar^i^/v/e/j/s 
in arc/($r fo /<far/7 fc> j'<9/ecf /he mo^f 
ooriren/'ent or7e. A/y/ai/j p/o/" /he pra^A 
of the {(%) as a ^hech on ^erct/c/s ^/)^n. 



5l> 

When a concrete problem calls for the 
cond'tfons cinder which ot certaiin Cj^uon- 
///y^Q, win he a MAX. or a MIN./if- is 
necessary first to note ivhort second 
c^uarttity ecu? be varied whose value 
will Control the value ot Q. This lot- 
ter /5 Ihe IIHDEP, MR in the problem 
and Q must be expressed w terms 
of it, Then Find the or\t,qr^s,Qncl 
proceed according to one of the three 
methods descr i bed. 

It f re^uenf/if happens that fhe formula 
obtain ed for Q is a power ^ a root^ 
or a reciprocal of some other auan- 
titif,^, which is easier to differen- 
tiate than Q. In such oi ease if- is 
advisable to final what valuer of 
the / note pen dent yartable make A/ a 
MAK, or a MIN, and then to inkr 
from these results what ore theMAK, 
and tllN, of Q. 

Example: Find the ^r^atest rectangle 
that can be inscribed in a circle. 

Solution: 
The circle must be regarded as a ^iven 




5/ 

one, and its rod) us will he a Constant, 
T{, The 3/ze [Area) of the recta n^f^ 
IS control fed bcj the length ta/re/f for 
one <3ide; then take one &ide as the 

indep. yon and call it ^ 
BycpresQ the other side 
in terms of x: il is 

^ ^fW^~C^^ 
The Area - which is the 
Cjuanti tLf to be max imiiQ<i 
— h: Q 's zx Vn^ - ( ^/k )* 
Bolh Q and its s<j^uare wili have a 
maximum value for the same -Xy and 

is a simpler quantitcf to Qxamine tor 
maxima and minima than Q, 
rut fr-t) s M - 4 R ^^- ■x\ 

Find cnlical arguments t)jf soly/n^: 

Crittcal -x's are -lo-o, %- ^n'Tl. 
The, values 0, Qnd-R^ Can be Ig- 
nored In soly/nf this problem. 
Examine, ^- F^fz ^Jf ^^« 3'''^ MtrHoo, 

V\R<x) = -'iblR\ ne^ahve^mditat-e^ & Ma*. 
.:-x*B^^;ve5 largest area square) xft 



58 

Maximum - M iNiMuM F^robl^ms 

It 15 not a ^uffic'ient solution of a 
problem in n^a/^imo c/nd minima to 
tind what critical arguments produce 
either: the Max. or MtN, values of the 
function should he clear I ij sloted 
together with the con d if ton that 
produces each^ The printed an Sixers 
to Stuch problems are abbrevia/'ed 
from the results that should be 
given in a proper solution, 

iNVESriGATE^ FOR MA)(/MA AND Ml NINA 

E.-I L^=: ( :k'-7x^-6;^ 

E"^ A beam of rectangular cross- 
sec tron is to be cut from a cjjTwdn- 



cqI \ocf 2^4 inches in diameter. 



5c? 
its 




itren^th is proportional to ^^^ 
■y^t^, its stiffness to %^(^, yL. 
if -x IS iis depth and y its breadth. 
Dimensions at strongest beam and ot 
stiffest beam thai can toe cut, rec^uired. 

E-? What ciflmder inscribed m 
given sphere has ^^^ ^ " 

greatest volume f 

Ans. T^ R r% 

H-S It the radius ot a sphere is 
12 inches, what is the 
hei^tit of a cone that 
can be turned down trom 
it with the least wa s te7 
Ans, 1^ inches, 

f-cy ThreQ thousand ^t^ gallon me as- 

ures with s(iuQre bases are ^^^ ^ 

to be made ot sheet copper. 

One gallon ^ Zdl cu, in, Wh at ^ 

are the best dimensions! <j,j4 K<j,i^t4ihl 

e-'IOFihd the least value of ic-*-^ 
A ns. ^ when X^i 




bo 



e-H WM 


value 


{east- and 


how 


Short will 


mat 


value iTialfe 


It? 



E-lZ In m airing q box out oF a 
square piece of tin the \ j-FTTTl, 
corners are cut out and la" : i 
wasted. Using i8" S(^uoires \ L j---- |- 
tiow shaJl one ^et greal-est cQpQc'itu 
for his mo/iet/ f 

Ans. Mal^e the box 3" tii^h. 

f -/3 The cost of coat per hour for 
drikin^ o steam ship t^arfes as the 
cube of the speed, dhow tt?at against 
a current of 4 mi. per hr, the ryjost 
economical sp^ed is hmt. per hn 

E-'4 A Mormon window consists of a 
rectcin^ie ^urmoctnted bcf a /^ ^ 
semi-circle , For a ^iven ( \ 

perimeter detetmine what 

wir)dow ^iil admit the mo%t 

light Ans, /^o/re the radius •= 

//? e perim eter ^Cnn-4) 



^1 

^-15 Hnd the alhtude and Volu^me. 
of^ the cone of iea^f- \/ol- 
ume ciroumscribGd about 

Arts. AJh 4n, Vol, f'^J^^. 




wjth 



gj 



E-lh In measuring resistance 

the percentage of 
error due to anjj 

error m sett/hj tH < c > 

slider, j&, \s inver^e/y proportional 
to Cac-a*). Show thot the best 
measurements can he mode when tht 
shder, ^, /s neor the middie of 3p(*ce o. 



t-ii What is itie greatest isosce/es 
trianQ/e fhof con be inscribed in 
a ^/Ve/7 circ/e ? ^^s^ ^(^uiioferai. 



E-ia Frhd fhe d/'mensions 
qreatest rectangle thof 
cqn b^ inscribed in a 
triQn^/e , &cich of whose, 
^fdos fs ^o inches in 
fen 3 1 hi 

Ans. 10 inches i 



th€. 




&,h5 incites. 




t-lH A rmitier i^ ho open 
from A to S. On a ^JT~^/' \; • " 

^eye/ through A ''a Q ^lih^^L ^_ 

SurMce of separahinn ' ''/ soofj-"' ^^A/- .' 
between a sott upp^r ;b ©///'' ' ' '" 
lotc/er through whicn a tunnel can he 
driven for ^\o a toot, and the 3oltd 
rock where the cost would be. ^30 
a toot. What stiould 1 1 cost P 

A^s, A boat ^IZ, 500^ 

t-ZO The hourly cost of fuel on a 
liner yanes as the, cube ofherspeod, 
bein^ ^^6 an hour tor a speed of lO 
mi .per hn A// o/her expenses come h 
^/4S cfti hour In what fCme should 
^she plan to moke, q 3ooo mile tn'p'( 
Ans. In &claif3 Ifhrs, 

t-Z\ Oi^i'dQ Q line I'nto four pQrts 
3 OS to make the. recta n^Je formed 
from them as (urge qs pos^il^fe, 
Ans^ ^ e fua/ parts. 

0-22 A box, s^UQre base and no 
coiner, /s to be made , usincf 
as httlt material as possible. 



^ h3 

]VhQt' di mansions should be u^ed 
for a CQpacf/t/ of fOdcu/'? d\ 6"x i?l 

E-Z3 In measurinj^ current wifh a 
Tangent Gali/anometer, the percenta^eof 
error due to Q smctll error m the 
teaclin^,^ %% /s proportional to 

(tan -^'^ -t- cot ^* ) 
dhokv that it /s least when :x'*-^S^ 
(CompQre tt?is prolbJem with B - 10.) 

E -2.4- A tofrmer warrts two e<facit 
rectangular ben-Jfords X side o-F bQrr\ 
eofCh to co^tof/n 6oo <boo^ \ boo • 
j£. ft Joking Qdi/an- ^.-%-XA%-\ 
ta^e. of one 5/de ot Me boirn, 
what /s the least cost of the Job 
ot 4z cents a Foot for fencing put upl 
Ans. ^5,4-0 

£"--35 The speed of waives 
length is X (lambda) 
is proportion ail to 



V 



a A 




Show that thecj ctdvance most 
^lowli^ w hen X -^ cl. 



i;4- 

H-Zb A holf-ton w&i^ht hanj^s, two 
ft, from the end of ct te^er z' 

and is ho be tcttsed btj ' ' \ 

lifting at p, if an iron I 6 ^ 

/eyer js to be used wef^fy/n^ lO/bs, 
fo the toof, what length of lever win 
make the easiest J/f/-? Zoft, 

t-ZX At what po/nt^ B, shall a 

passenqer jump from — - 

his car, which ooes ^x \^ 
at tf?e rate of &m/, ''X^|y*. 

per hr^ that he maj^ ^H 

reach c (S/s quiclxljj gis possible? 
he can walH 4-mHes per hn 

Ans, Mcfke AB^ il. 55ij(^s, 

B~2d Through the point (oijb) a line, 
as short as possible, 
/s draurn from one ax- '^ep-'^^ 

/s fo the other, 6 how 



that the slope of this line must 
be -.yW" and its length (ah-h b^;^ 

B --z^ in the problem of the flow of 
air through a small open in g^ it is 
desired to get tne moixlniurn 



b5 
value for -x,^^ ^ -t. '^ }f (jfarnma) 

E-30 The, power ^f^^n to an exter- 
nal cfrcuU t>y a ZOVo/t generator of 
mternaJ tesistatice iSohms when the. 
current is -i amperes is [zOi-lSil 
watts. With what current can ttiis 
(generator deliver the most power "^ 
Ans, S5 ^ wal-ts with 5 ^ amperes, 

^-3/ /f the leih' of reflection were this: 
The path from A to B> bjj way m-^^;- — H 
of the rvirror^ Mm\ must be the yil^^^^ 
shortest possible one'^ show ^-^ ^^ 
that ft follows that- the an^je of inci- 
d&nee,a,niust e^ual the^ Qti^le of re- 
flection,^. Use a case in which A ond 
& are ofi/fc/ista/Tl from tiMl 

t-3Z A ^QS-holder is a cylinder closed 
at the tippet end and open at 
the k^ottom whtre it sinks into 
water. What are the most' 
economical dimensions for 
the cylinder? 

Ans, radius - he iyht. 



H-33 For Q certain sum o man agrees 
to build a- rectangular r^-'^l ^ ^^ 
\vQteH' toink and /the ft - \ L ^^ 

witti lead. The hase is to he q s^utfte 
and it is to hold 3o00 cu>Peet, 
Wliat dimensions ivill mcitre the 
cosh of /ihinq teqst ? 

Ans. V fh X l&.Z Ft. X 16.2 ft 



B -34 A teleorQph pole at a jbend in 
the road is protected from 
tipping o/er bj/ q zoff. staj^ 
fastened ho the pole and 
to a st aire » How hpr hrom 



the pole should the shake be driven 
in order that the tension in the 
stQi^-wire mcfcf ha ire the ^reefh&st mo- 
ment about the foot of the pelet 
Ans, /4, 14- ft 

E -35 /^ man in a row boot is 3mi. 
from the nearest point, A, of a stra/fht 
beach. He i4^ishes to reach Q point 
on the beaoh 6 mi. from A, he coin 
roiv 3 rni per hr, and walk 4mi\per 
hr. How -s ho// he row'> 

Ans. 3o as ho watic J*6( miles. 



r- 36) What number exoe&cfs /As 
square b(j the greatest amounf^, 

Ans. One-hQif. 

E-37 i ^tsh to maHe the most- 
capacious box J can prom \ l 
a piece of aardlooard 1 I 

3^" X 14". tfow bi^ ot square ^ ' 

6ha// / cut trom each oorner he tore 

toldinq up? 

Ans. 3 in. sc^uares. 

E-3a For wtiat breadth across the 
Top has this fi^i^r^, ^----?-— -- 
summetrtcal and ^fn, "^\_±__J^ 
on each ^iren side, the greatest arec^^ 

Ans. S inches. 

E-39 What h the lar^&st Box mail- 
able in Enqlarrd, i^/liere the regula- 
tions forbid mai/tns ff ^he sum of 
length and girth exceeds six teet( 
^ ^ Ans. /thx/ft.x 2 ft, 

B-40 What 13 l-he iofrjesh c(/findrical 
pacKoge malleable in Bn^le^nd? 

Ans. Ztr. lon^, ^ft, circum. 



48 

Integration 

The process o F Nndin^ the di'fFeren- 
tial t one function leads to q s ee- 
ond tunch'on. The process ot nstrac- 
rn^ /y?/5 process, trom the differen- 
tial to the fu no hi on on^ I nail if dif- 
ferenliateo/j is called J NTBGRATtoN, 
and it is indiccfted bt^ jplacinq the 
Inteqral SiGNy J" , {an old -fashioned 
lon^-s), before the differential to 
be fntej^ rated ^ Therefore if 

d f=(-x:) s q) r-^)c3<x 

then c 

But fW is not the onlj^ ^uctntittf 
whose differential is q>i^) d%, for 
if A is a/7cf constant whatei^er; 

The number^ A, is called a CoNSTAm 
or Integration^ and must be remern- 
bered as an essential part of eveiy 
result obtained bi^ integration. 



^7 

domehm^s the. ar blttarif constant^ 
[like the exponent one),i% *'undQr stood" 
and not- i^ritten, huh this /ibettcf 
the student should not perm it 
hi'mse/t at fh/3 starve of >?/s studies. 

The hre^ofn^ DeriNntoN of iNJEGFfA- 
7 1 ON hiQj^ he presented in this form: 
J dix =. tt -f- A 

F/Ve formulcis for mte^rafion follow 
from the five for mu/as For differen- 
tia/Sj which for this purpose <^re more 
con/en)entlu put into the. forms: 

die r^ o 

ciCxt + v) = dix -♦• dv 

ci(cLt) — c d\i 

d(.\>TM) - xt dv — x> du 

These ore now to be chornqed to the 
fhtegal form, mctitin^ use of the ictwi 

J* du ^ \L -^ A. 
Note fhat if n=-/, ttiere is no such 
quantitij as t£:""-rCn*}) to differentiate. 



-JO 

So =^ c^ A' ^A 

JccSw.^ o-uu-t-A - c ] d^u 

Su'^du^ y^ ^A f Provided yv 15 

These Form 14/ a$ may he restated in 
botter form /^r use as ?olhw&: 

1 So-A.y^doictn Arbitary Constant 
to eve^rcf thte^ral. 

n S(cUi-Klv)s S du-i- J dxf, tntegrahe the 
Terms of- a Sum se/^ara/'e/i^ , 

m Scdu ^ c Sdu. A Constant Faotoj^ rmif 
he moved from one sfde of the 
J-si^n to the. other, 

IT Jt/du - -oSdu' JxLdv. A Variable Factor 
may not be moved poist the 
J-si^n without introducing 01 
certain neyt/ fnte^rat as an 
offset. 

'S Su^du = V'.^'^' ^ ^j^ ~Proyidedi n^-u 
""■** ' iNCReAse the 

exponent of the jbase by one and 
oiv/pe btf new exponent, unless 
the icftter can7&s out =• ■z.ero. 



7/ 
The use ofji 13 tr&aled on pci^e 150, 

In usin^ X ma he sure hhah the Power 
(s multiplied btf the dftferential of its 
base. To ' proi^ide for this a constoint 
factor mau often be introduced ctnd 
offset' thus, J(z-5 %^ f'ic^d% 

To yerift^ an example Jh inte^ratm 
differentiate the resu/t^ and obtain the 
\UTEQnj\NDy or c^uantitu original I lj offect- 
ed bif the sf^n or integration, 

Example I J(^%^ -r m -^l-x^-t-i] -x,) d?s 

Verification: 

= the integrand. 
hloTE tt?athiclxSt)d,%^Z Jdo^-f-pc) d.p 



OffiLL ON Integrals 



Tivo I'nte^rQ^s of- the same m he ^ rand 
tncicf differ m hhe Form oF the cort- 
shant of inte^rcition ivhen worked hj/ 
differerit methods^ Thus Sl^^O^ d%- 



--^^^•i-x^i'Xi'ir -^A 



r " ^ ■ ■ ^ - 

or « Ji%^+zvc-f dx== -^-x.^ ^ ^ -i- ^ -t A 

These results AgR£B in m&cininj^: 

^X^-f -yc^-K -f Soj^ti UndBTBRMINEO CONSTANl 
iNTeGRAT^: \/£RlFY RBSULT fN EACH CASBl 

(i^-t } ^ die F-7 J Ui^r 
f'-d Ji^-h dx) F-a JUnxK I) dm 

f'4 I -fsTT cis F-? i (i -(J ) cl(2. 

F-5 JVi^ di f'lO J sVi^ ds 

f-n Hzx^^rx -if^ -^-7:^"^) d% 



73 

since iNTEGRAl-ION 15 the BtVBHS^ of 

DtFF^R£NTiAri»N^ questions like- th^ pr^ 
cedmj mc(^ k>e. ashed in these. 3 
dIffUrent hrms: 



74 

Determination or CoNsr/\Nr 

The constonf of infe^i^at/'on fs orb/'- 
ttary onc^ i/n/f/70w/7 so /ar os the tip- 
i<'ers(jt of^ a Gf/fferenfiaf/on js con- 
cerneot. Buf some foot oboof the 
i/a/i/es of the yor/ah/es /nk'oti^ec/ 
rvau /?7(?/(rs /t poss^ih/e to c^etermme 
the ni/mcn'ca/ ya/c/e of the cons'^nt 

Example: /I boo^c^ /'s cfroppeo/ from a 
height of ROf/: /ts- speect /s 3%ftp€r 
sec. t^hies the /lumber of sec. ///7^ 
O forma/a for /'/s he/fht /h terms 
of The t/me. 

Sofc/t/}>r?: let t sec. — t/m^ of fr/J/r?^ 
^^^'' to a he/'^ht of x ft 

Downward speeo^'^ ^t' 
f^er7cc -i^^^^t 
or c/^=^ — 3a t ctt 






1 



Infearat/n^ X= fd'?i = J-3Zt dt 

}fhe// t yyas zero j ;pc yvayp ZOy hence 
ZO =z-l6xO^ 4-A 

A =^0 

Aooorcf/h^/y /?e/fht ^ (zo ~ 161^) /f. 



75 



Megrate ancf c/etermine the consr^nr hy 
rrii^af^^ of ffye co/?Ci'Jtionf ^/vep /r? eac/? case. 

f-Z4 W= J :^Vx^tJ (i% Ginc/ yfhen x-^. 



f6 

Integral Rate Problems. 

/Is problems ih n^Te of j/7cn$ase> 
werr^ soli/ec/ by c^/Y/ere/jf/af'/or; w/j^n ffje , 
R/4TE (^speeciljOcc^e/^raf/b/7jt^ens/f(/, force^ 
pressunfj^/c.^ s^e pp. 33, is 8) yyas refi/Zrecii 
sp i/y/?e/7 ^ Rate /s ^jVe/? the problem 
is solyec/ bj/ /?7^a/7^ cf /Wte^raf/'o/?. 

£x^/77p/e: The pressure 0/7 a a/a/n off 
oa(/ ^epf/? is 6^ i /bs: per sf. ff> 
X f/pe c/ejpf/? //7 /eet. l/l/jhaf" /^s^ the 
l^/?o/e pressc^n^ o/? C7 c/a/?7 M'yy/e/e^ 8Weep7 
The c/rt^rap^ pn^ssure or; 
o /por/zonfcsr/ s'/'rip y^ 
d Force -f ^Afrea 
Her?C€ , /he exact ^ pressure 

Tak/hp A Sf, ft or t^e a rea (Var) a/ow/? 

fo c/ept/? k ff{i^= /nO£P. Var.) we gef 

/l= ZOfy. whence dA= ^O dh 

:.F== i IZSOf^- -h a cons font 
Whe/r ;?=0, f=^o, hence the const— o 

For fhe who/e c/a/rr, ^=8 amal Force is 
GlS>^8'^ /6s, or ^OTons. 




77 



Q-\. 7?ie speec/ of a /baa/c/ that j^/at/s- 
/>W %=5*//f c?f /^(^ //me t-^^ sec. /s 
Si(lft) ft per sec. Work ^^Z" /^^ /or- 
7r7o/a for k //7 /erms of t. 

» 

G-R TTpe speech farmu/a he/n^ 37^^'^ 
ftper^sec, P7€ Soaf(/ rec?ch/h9 ^= /^ /^ 
whe/7 t^ ^tsecj ///7^ ^/^^^ P^ ^^^ 
yvhen t y^^*^ zero. /^/7^. 8/^ /f 

G-3 /^/^ e/^s/>e iba//oon /^ he/n^ fi//eG/ 
witf? <7as. /f re/77a//7sr spher/ca/, /-^ 
rac/zW r ff., //rcreas/>7f af of ra/e 

m be/rjQ the vumber of mmu/es s/rrce 
r kvas^ero, /^oyy /or?^ cfoes/f/a/fc 
fo /h//a/e // fo c/ cf/a/Tye/ler or 
RO ffP /I/7S. /Iboi^f ^5 mipcjfes. 

g - 4 (jiu^ n-'^y-, a/7c/i^hefj x^j ^=o. ^hat 
/'s y /h Ter/vs of ^/^ /lns,Lj=:x%z(}-^) 

Q,'^ Acce/eraf/cn be/h^ c^/7sfar7for7cf^eed 
ancf c//s/. ibe/r?^ zero yvhen ffwe /s zero, 
proi^c c^f'sTffnce yar/es crs Sf. of //mc. 



78 

G'6 AtH o'c/oo/r Q j's /hcreas'/nf c^t fhe 
r<3'fe of (^!r*+T+9 ^^^'^P-^^ /'^ i^a/i^e 
/s ^o ur?/^ e7f 3 o'c/ocM yV/raf' /> /Xr 
yo/ue af oc^ar/^r pa^t four f 
/}/?s. /ibaut 76 i^n/U. 

G"! n /?7//7€ /^ c/eeper?^^ of the 
rafe of (zo-^^^ ,o\5ff. /per- year^ u 
J)e/r?y the nc/vber of (/ears s/nce 
the /77//7e i/ya^ ape/7ea/, ftoyv c/eep 
/s // c/f f/?e ^ncf of 15 yearsC 
Ans. 5^8. feet 

(3-8 Tf?c cross-sec^ 0/7 of a r/Ver //7. 
creates a f f/je ro/e of (76o(/f(io^^^) 
sf. ft p. /v//e ap X /77//es /^rom the 
>S'ource, yVhat /s ;J^^ cross- S(^ct/'o/7 at^ 
the rryout/? of <7 r/y<^r /ooo /77/tes' to/70? 
/lr7S. t/^coo. s^. ft, 

G-^ r/?C /ror/zo/z/^/ cro ss- sect'/ 0/7 s 
of o hot/ are Q/yen 6l/ Ihe /or/77(//cf: 

S=l{7.oo-kf - \6.fl sf. ft: 
^ be/r?^ the ot/j-f: />etoyy tfie y/o/er- 
t/ne, C(7/cu/c/fe tota/ c//'sp/c?c€- 
r/7e/7t' /f sf7/p ctrayyj" 30 /f: 
A/7S. A/hoe^t ^C/77/ttJc>/? cc^. ft! 



G-fO» Sh^i4^ f/yaf the cr^s-s-sec^'c>r7 af 
a r/'^h/" 6!^/?^^ /id/yhT H en?., 

ft err/. /?e/oi/ir f/ie yer/ex, /^ 

7t ^ R*" -r W*" Sf. crrr. 

Then 61/ //7/e ^/-aZ/'c}/? pvor/r o^t //he 
i^o////77e of f/ze ca/7e. 

&"■] I /It nc^o/? tt/e t/'c/es //ei^t/t' m/as 
6 ft aSok^e mear; sea /ei^e/; 7n /^/nc/7es 
Jater ft mat fi^/t/Wp a/^ 79? e n^/c of 
[tc -^ 30nf 60f 10-^2 />?• p. /w/7. f/hcf /t$ 
hei^fyf (7r [9.'^o RM. Ans. 5' 1^" 

G-l^ The acce/erGf/70/7 of a /neteor 
/s /hi/'d'rse/i/ /7S /^e ^f, of /fs <^/st 
fro/7? t/?e center of the e/nr/h, he/hp 
5A ftp. vf/?. s. (^or Ttr nr/ \ per sec. per scaj 
c^f tt/e ^^(yrfacef ^000 m/'. fr///n the 
cer/fer t/s spee^ /yc^s ^ /n/'.p. s. 3000 
777 t at^o^e the st^rface y i<y/Th yvh^f^ 
^peecf c/o<$s tt s'/ft/fe^ ^ 

/^77S. At?oc/t'^k /77/ per sec. 
Uotb: 7a Jrrte^rafe 4^^=-— ^/^s/?oi^ 
tZ/i^f irdt = c/x onar nrc/Zf/pty cop- 
re spon(^frFp J'tctes of these livo e^cA?- 
t/'ons. Ttre/7 /htep'ra/e ^e ira^ir-^yc^c^K 



80 

G-ld For (? frafi//?^ Soaf^, x ff. aboye, 

ureaf //? j^econc/s, // n/y&n t^o the 
spee^ is zero a/ra^ //7e /ye/phr Js V\ 
ft abore' ^rou/7a(, s/?/pi^ by J/7te^ra//'nf 
onct <^eterm/m/7^ consfo/rfi' that' ^r 
<7rry //rsta/rf aerofec/ 3^ t seco/rc^s 
r speed downworc/^ 39>.Zi ft per sec. 
\he/^/Tfabor(ffroor7ct--(W-\bA t) ft. 

6-14 Dur/np or? exp/osm the pos /rr 
O cy///7ct^r /s 6/o//7^ i^ork bu pc/s/?- 
tny 0/7 //?e p/s/^/? af /he rc7re of 

[60,000, — M^[0^(i''.osy] ft /h per sec, 
Qt t sec. s/f?ce the spar A s/ar/ect /he 
exp/os/m. ffoyv mcch yvork ts c/a/ye 
op(7//7sf' the p/sfor^ rh the first /d sec.? 
/Irs. ^000 ft /J^s, 

6-15 Thie s/op^ of a cc^rve e<^i/ofs 
the S(^cyc^re of the orcf/haie , x, a/?£/ 
fhe po/r't r^y 7) /s on the 
ci/rrt I f/ha /ts ef(y at/or?. 
/Ins. 3u^=- -x^ ^ \3. 

G -16 dra/res C7rc ^ra^c/a//u c^p/tp/xa/ 
so thc^t c? tr^//7 Jvd^/ys onpkv/^ of" 



en rafe propor^h/7a/ fcf ffye s'(^£yard'^ 
P?C t//77e e/crpsec/ >p//7C€ o/pp/zcGr/'/cir? 
hepa/P. yy^^m oc^f /vrn?i//a for c//star7ce 
moKtSc/zh first t secor?c/s, ana/ af^/^r^ 
mi/re t/?(^ propor/ima/ify -^aofor /fa 
fra//7 ^o/'/79 50/??/. p. /ir can be stop- 
pec/ ff7 w/s /vay • /p ^5 s^^c. 

/I/7S. J. 55 y 6//7//S be/hf J77/'." sec . 

G^IJ fbr a bo// han^/rrp by a spnhp 
a/!?c/ K/j^ra^h^ ^ =" - /e*^, v- be- c^ \ 
/n^ //?€ speac/ /n f//CLJ)i^c., ^ be- ^J 
/n^ //?e c//s/. j'/?oin^r/ //? fee// ^^ 
arr?c/ t b^ez/Tf a /?a/?7^er of sec, Q^ f 
Shoi/y f/?cff vo/t-du arro/ //?te^ra/e as 
/r? G~l^. Sfypi/y thor V — /#«/ a^-^^ , 
a (7/7cf /# be//7^ co/rs/orr/s. 

(j^lB The far/77(^/c^ for fhe force /yeci/ 
ec/ fo ra/se a/7 /fya/ra'0'//c e/ifra/or %//$ 

F//7c/ fhe i/vor/r c/o/ve //? ra/ir/a^ f/?e 
e/ era/or Soff /Iboc^f HZi ft. -tons. 

F/r?c/ -f(^ /f ' //s a^er/i/crf/'i^e/s 




G -1^0 /4ssc^/77///f Pjat the /a/u(S' of ct 
r?/o/70^c7/7y /ree c^irer ^Ot/rs. o/af, sat/ 
u If ears o/^, /hcr^(^Si^s if Jeff fa 
qroir/ of 0/7 c^rrrnuaf rafe of 

fine/ fhe fncr^ase, //7 k't^fue from 
fhe aye of too years fo the ape 
of zoo y<^or^. 

/}/7S. /ifpoc^r ^8, 000. 

G-^l /4 fpocfcf foff//7f c/oyv/7 a f?^e fn>f?7 
surface to ce/ifer of earth, arrives with 
what speecf, if /fs speeaf fs Trff.p.s: 
on reaching x n?/, befot^ fhe sc^rfoce 
wfyere i/-^ - CsTs . - . 008%) 52sBo ? 
/^/7s, ^4* f ^ n?/fes per sec. 

G-M The pat/-roff of a orcsperous con- 
cert? fncreoses afa/fi/j ar fhe rafe ^ 
^{lO'^-^lo^) per uear a/7nua//i/y ^ 6e/npW?e 
VO, of t/fS. offer Jo^ /, ii[03, ffrfcf hfaf 
amf. waoes pa/ of oti/rfn^ fheffe<^r \^0% 
/Irjs, ^ 10,^0^.00 

/fs- speec/ 

^p. seep. /ach. ff if sfr/Mes' mrffh Of 




85 

^peed of Z4oo ff.p, <pec. Aoiv /hr i/yil/ 
it penefrafis, x Jbi^/h^ the pa of /hoA^s 

/I m. /^Soc^t '^'^ inches', 

G^T.^ /f ar? iror7 6a// fa//s /h i/^^a/er 
wi/h /fa/f /he occe/era/zpt? if fias /n 
air (the /a/fer /s d^.^ ft p.^f,s,\ how 
/on^ c/oej^ if fi7/fe to fa// rrom c; 
h^/^ht of 1000 ff. a/?oy€ yva/er t:> a 
depth of 500 ff, /?e/oi^ its sorface? 
4/7S, About ^^ sec, 

6-^5 The ra^/cfs of a j-phere /hcreases 
of a ra/e fpirerse/y proporflonal to ih 
opvn /er7fth. O^t a for/ni/Za for tie yo/- 
u/ve //7 terrvs of t'/?7e, V^HCti^^'*- 

6-^6 /f7 pi///i/7f a stcthi^ cc/t of fhe 
^rpc^/7it, the res/s/ance^ B Ihs^ o/ecreases 
OS t/ie sta/re ^/yes^ so that L, /he /7c. 
of /hches it has heer? rafst^^;^, fs re- 
/ote^ to H occorc/ing tc the formula', 

Ca/cc'/at^ the krorh a/o/7e /n ra/sjrii^ 
/he stahe t/ie ffrst f/ve jhches, 
At7s. ;i7.8 ft 26s. 



84 

TFtANSCJ^NOENTAL FUNCTIONS, 

The fc^/7c//o/7s yw/?/a/? /lat/e been meed- 
ed fhi/s far w fh/s ^o^/r hare a// 
bee/7 /orr?7e(!/ frprrj the ar^iyrnetit hy 
comhfnat/ons of aMh^, ^uJbfracTin^, 
/77c//f/pJyj/7^, ^///W/h^, ra/$/nj to inte^tvl 
fracf/ona/^ orne^afi/^ consfg/?f pojn^ers. 
/t fc/r7ct/or7 /rWcf(? format/o/7 oa//s for 
an^ ot/rer processes fhar ^ c/ef/nite 
set of th^se /> oa/fect Tff/^/M sc en den- 
T/^L. TTie /posf fa/77/ /far fra/7scenc/e/7- 
Taf fc//7ot/'o/7s are the fr/foncmetrfc 
fc^nct/ans, tt/e c/rci^/ar (or inverse 
frf^arJO/vefric) fi/actfa/is^ fhe/o^r/thm^ 
(/net tt7e/r frkifrses, the eicponeryf/d/ 
fi/nct/ors fhase yv/th vanahte exponenfj 

The transcenatenta/ fi//7cf/h/7s /trafare 
777ost^ co/7Sp/bc/pus /h e/em. co/ci/tas are\ 

Si/Z TT 

COS ir 

ta.n ir 
arc s/n w 
arc tan i^ 



as 

The s/x steps c/escr/hec/ cr? paqe 16 
/y/^ 6e /b/Ui^ec/ in c/ea^c/oina the 
Cifi'ffere^'tia/s af the fancfions'. 

s/n I/- 
and from these re^ri/Jts the c^/f/eren- 
t/'ff/s of t/?e othsr ffVe yyl/l ffow. 

Fut ^ s lc>r w 

(5) ^y= Zo^Q/'-hAu-) — loair 



(6) du = 



^dLao^v)^irlo^e 



//m/t(see pa9e Sj /«r /he rK/mbcr- 
X.'\\S?.S.„.,kvhich ts ca//e<f 6. 



56 

O) /dz= tr//2 (v-hA v) - nn v 

=i?.c0s{(^v^A>t) . sin iA V 

(A) AZ.=, cosfv 4^^\ > i^. ^^ i ^^ 



Csr i2_ ^^^j^ .^. gx 



^AV 



If looy" /s a Cont^OH LogariwMj (Base tO\ 
orJi^ P?e o/7$^/e k /s SKprcssecs/ in 
De6RHes> then fhe nvo Multipliers 

AV-0 LAY ^^0 /I ^y/j/^^ 

///n/f' Cree pa^e 5) is the pc/mber 

the number n 6e/n^ such that n of the 
Units /'/? yyf?/ct? the /4m&le /s measi/red 
=* 1 H^voL\)f\oH = ^71 i?AOi/ws = '3(>o \ Therefore 
the MuLTi PL/ ER, «it -r n ^ /77^^/ be the 
F&40MN-EoL>iw4LBNT a/ ;^(f /Ingle-Unit ^J"^^ 



87 
lo^6 ar?c/ ^itfri 
hat^t^ i/7coni/en/ent rn/mcrical ^a/ues\ 
.^S^Z^'I^ one/ .o/r^S'S-h 

di/t if M4PERI/4N (<?/- NatuR/Al) /ofarilhms 
(Bas€yC^2^Sl\&^) ont^ ^AO\AHS ore useci 
/hese mi/ffipJiers take their simpUst 
an^ piosf conyen/ent i/^cf/ues t ^a/7ie/i/, 
each is efi/a/ fb the f54 croR ONE <^ncl 
$0 neect ryot 6e ^yr///e/7 //?. 

fbr ftiis teas 0/7 C- Loca r i thm 5 a net 
R/4DMH-MeasvR£ ^re usee/ i/\fher^yer 
ct/f/er€nt7a//cr7 or /hte^raf/'or? /s /h- 
yot^ect. Tah/es- ^afw/// fac/t/tate ft?e 
use of t/jese s^ sterns are g/^en orr 
papes 154- 15 5 • Tfjroo^trocf t/?e re- 
777(?//7ater ef f/7/s fext' /'t n/ust be 
(ynGtersfoac/ /'/hcff /opant/7/77S c^re to 
f/?€ BA^t 6 unless seme other hase 
/s mct/cafeetj one/ /haf e>/t arr^/es 
are eApr&sseat /h f^ADiANS untess 
some other u/7/t ts inet/ca/eet. 

Err d sin 6"^^ ^=^ cof6 d6 



88 

Frorv yl one/ HL ore obTaineo/', 



xc* d arc smu^ 



du 



v'/- w^ 



^d arctanu^ / f* j ^ 



zni PuTu'^ 6^ TTien 2o^^z=. u 2oo € ^u 
du =r u du ^ 

= lCl-sin''Of^(-Uin6)cosSd^ 
^ — sin 6 da 



mcor? ^e^ Momber of f?/^0MN5 //? ihean^/e 
w^ose sine er Ta/i^enf /s u . Tlhet/ do 
not /nea/7 Ppe any/es the/vse/i/es and 
therefore may nercr be e}(pn^ss€d 
as so ma/iy decrees. 



a? 

^ cos&dO' (cos 9) ^ 
-= [I -h tan'-SldO 

XL Fufd^ arc-s/pu^ T'hen u- sin 9 

dcL = cos9 clB 

*^ COS P 

d arC'Sin u == j/J^i^%^ 

xn Put<B^ aro-tan u. Then u - tan <p 

oCa^ sec*(pd^ 

_ cCu 



d arc-tan u ^^ j-h u'^ 



F(Prmu/a^ for the d/fferenffcfts of a, 
sec&^ art-cos u, etc.^ are aften g/i^en 
b(/t fhec/ are not often refu/red W 
t^cn are east/^ yvorked ai^t i)u the 
mettrods used for nn, iL,QnJ SL. 



^0 

Drill on Transcendent a ls 

Formu/cfs m to ~sn are. cony^rhed mh 
the inte^r<f^ form below. Observe that 
sa provides for A^e ca^e. to which 
exception is lYiade under 3r. 

// n=^ o if n^--t 

d sin'u = cos u du yh jcosv^d'U -sinu^A 
d cos u = -sin u da B Jstn u du= -cos tt+A 
d Qre-sin tc = t^= it fr== - art- sin uM 
The defif^ih'on of to^aritiim is rec ailed: 

If A/ S ©^ 



H-t) JC05 3-3K, <i% 

u n f -J^ 
H-8 1 






H-Z d cos e3 
H-3 d log 5irA^ 

H-5 d arc tan ^ 

H-19 d lo^y^ilLi*! (spill- up t'rsi) 
H-/5 / e~^ d;i: 



^ f Ar - dr> 



H -^i d arc sec it 



H-17 /ii§^a^ H-;t3 d cote 

h'iS J d sec 9 



j^-lS d to 



^ 



M-^6 d siTv^x*^!;! 



10 



<?2 

Miscellaneous Pro&l^ms 

rl Air /s beJn^ compressed I'n 
closed oj/liyid&t-, T/7e /T 
outside cfir heep^ a a — ^ 



"Tvr 






pressure of 15 lbs, per u 
5^. in. on one ^rde of- the pishon 
atid the compressed oiir res/sfs with a 
pressure or 15 Ibs.pspln. xfhe quo- 
tie hi- of the on'^inal irolumi^ divided hjf 
ttje fhst-cinhcinQou3 volume. lYhat w&rk 
must' be done ho reduce Ihcu.ln.of 
a if ho i its orip'ncnl \/oiurYt€p,'tht 
piston 01 r ecu is '4sp in. 

Ar\S. 7Zi it?ch lbs. 

I-^ Weber showed that the number 
of heat^units ne(iesscir(/ ho taise one 
^rorn ot diamond from O^Coj.ta l''C^. is 

,0q4*i I -i- ,000 4CJ1 i^— . 000 OOO 12.1^ 

Find the specific heal- [ heat-units p^r 

decree rise In temperature. ) ot the 

ordinart^ room tempercihure^ ZZ C^. 

Ans, .Hi>Y 

1-3 Accord rn^ t(f Newton's law of 
coo tin Of the temp, of a hot bod. 



93 
f^alls at a rahe proporhonai to its ab- 
solute t&mps How lon^j then, does 

it take ^ hodLf to cool from 5ooo^ 
AhSs to 1000* Abs, // it he^/'ns to 
cool o^ Me torte of 10" per sec.'? 
Ans. 13 rrtin. Z4i 5ec. 

1-4 If the aeceieration of a Jbodu 
varies d/r&etiif as ifs speed, shaiv 
t/70f/' both speed and distance are 
QixponentiaJ functions of fhe ffme, 

1-5 A vapor under conatant pressure 
IS heln^ used as a thermometer, ^ ^ 
If it expands at a rctte per 
deyree rise in fenjp, Wht'ch is> 

inverseiif proportional to Its vol- 



ume j and if its volume would be 
zero at Abs,Zcro, show that the temp, 
will be diroctlLf proport/onoil to Ma 
S(^aare of the volume of the v^ipoc 

1-6 The speed of o falling boScf /s 
'<4.2,7'\/K" cm.p.sec.j fi en?, bem^ the, 
distance it has fallen. Showth&t 
the acceleration is const an/, and 
is eouol to 980 cm.p,sec^psec. 



CJ4 

1-7 A s/frra/ ca/r be ^taf7smffec/ bt/ 
a scfbmanr?if cab/<s qrnf cm. in cZ/i^- d 
meter^ /hsu/oTeaf with on mire/ope 1 
of Q(/tta percha t cm. fhick, of a speec/ ^ 

kf7(? meters per sec, l/yhaf j's the max- 
/rnu/n speec/ of ^i^na/tnf, o/iff for 
what m/ckness of //Js^uJaf/on? 
Aps!, I IZ3, 000, fo?i.p sec, L6^8 c/r?. 

I'S ^ ^(y'Gfye records ^e Aeffht,uf/^ 
of the t)'(fe /h tf^e form of a our/e that 
from II AM, to ^ Eh. satis fks t/?€ e(^c/otm 

^= x*'— v^x +-^ 
-X /be//?^ t/me after /?oon in /7oors, 
orjaf a hein^ measure oi up fr/>m 
Q cerraip mark. Arra/f^e ^ t^f^te to 
shpkv the rate at y^/rich the tWe 
rises or ebbs at t/?t ey'en fiaif- 
/rours fro/n II AH fo ^.RM. jhc/c/'sire. 

l"^ A yvhee/^ ra^/c/s H ft, in^iihjfxecf 
center^ turris at tt^e / Ni • 
rate ^ of ^ (omeQa) ( y<^fi\\ 
raoiians p, sec. iYork ( ^J--'4 
opt o forp?c^ia fc^ ^^^ \ J 

Height (aboi^e a iei^el ^ ^ 



15 

clrtii/V/7 /^t£^(y^h the huh) of a dot on 
Tht rin? of ffi€' yuhee/ - ih terms 
of the on^/f 6, - on/;/ fro/n it cr 
farm (//a for the ^peecf of rfs/h^, 

I-IO yyortr 01/ 1 a formt//a for the s/cfe- 
ways moT/'on pact j/oeect of a /bi/^ 
that /^^f^f^, ^^ ftie r/'m of tf?f t^tieel 
ctesaritei^ /r? I-f , at a point jWtthen 
te^el y)^/fh ttie hi^hj onet craw/s a- 
for?f cf ^poke at the rate w ftp. 'Sec. 

I-i; /n th/s ra/t'-sp/'e^derAMhHMf 

scre^ ^hortfnesH^ iff?, /us. &^S£^§-^^ 
Ati^/f/^t rate are ffje ^ ><^ — 
raiU 9preacf/h^ ? Ans, s fn, psec. 

I-IZ h f/hct ff7€ octoa/ Hit force ^f^ 
senct^ the hoat a/feacL 
^et f^-e component of^ _wj 
yv/kof-Pr^s^ore /lorma/ \ 
to t/7€ ^n//y, tf^err f/hcf ff" 
tfre component of tt?/s y ^ 

a/on^ t/?e cf/'rect/on of tfie /(eH Hov^ 
fyffi/ t/f€ so // best J^ sef^anof i/vh^P^ 
fins. So ffrot x''-9o*'^i^'*hecac^seL.,.. 




96 

1-/3 From the o/afa ^rVe/? //7 1-8 
f/hcf f/if^ ^'rr7€ (fp fh€ ^^arest m/nuff) 

I- /4 /f pa^K on^ uc/p cfre e^i^l 
shc>iv thaf' on cf p-y-ct/a9fani^ 



I- 1 5 A coun/er-s/h/f hoi^s cut a conk- 
a/ ho/^' i/vhose a/i^/e /'s ^o* r^^^^^ 
/f f/f-e^ art^a c?f we ccn/ca/ i \\/ % 
S6/rf0C-e /n creases i/rr/'/ortn/f/^ L — '* y 
shoi/v fhaf the c/epTh increases at 
a rate /n^etse/^/ os tht afepfh , cfna/ 
^c yo/, of ara/e cfj'recfitf as tire o/^/h, 

I-/6 The exp/i^s/or? of a /vef^^r ^ert/^s 
oc/t a spher/'ca/ /)^aye, yyh^se si/rface 
oc/ygnces of a rafe of 1500 ft p i$c. 
/h a/f ^ free ff 0ns. ^f i^hcyf rafe fs 
fhe ro/ome' of afr af/sfvrf>ei^ 
/ncreasf/^f offer 10 sec (" 

Ans. -f^ ^ /O"" cu ff. per sec. 




9T 

J -1 7 Force in c^i/nes afc^als /rrasf in 
g^rams a acce/eraf'/on /h cm. per sec 
per sec. Dec/i/ce a fi:>r/77i//a for cen- 
Tripefc^/ force /h fhe /of/ow/hf monner. 
Get th^ form u fa for 
vertic(7f height of a 
pc>fnf ^ m^i/fn^ arc>i//7ai 
a .. o/rc/t^f raa//i/s K a/n^ 
^ffh o s/peeof of u- c/77. 
per sec. Fro/77 fli/s 
fin^ the v-^rffcc^/ cpmponenf o;f 
speec/ anc/ fhe \/erffca/ acce/eraff^n. 
The acceferatfon fc>\<\/artpf fhe center 
mat/ i^e founcf tn/ fafffr?^ fhe yer^fcaf 
cfcceferaffer? of ffie fnS'Tant fhe i,/voi^- 
in^ po/hf fs af fhe i^otfom c^f ffiewheet 

I~/8 ]^he/7 a ch//y is pfececf in a eiP^r- 
renf o/rcf refea^seef, /r sJarfs frorr? 
resf (7/7cf receii^es o-r/ occefi$raf/6/7 
proporffonaf fo f/ie c/fffere/7ce between 
its oyvn speecf c/npf //?af of fh€ cur- 
rent, ff if7e s/aeecf ^/ ^e correct f^ 
&/v/fes on fc>i/r^ ancf offer haff^ sec- 
ond fhe ch//^ fs ffoinp 6 nj/fes an foot, 
sf?ot^ fh^f fhe proporf/'onafjf^ fbofi^r 
I's ^V[% ^^ ^^/^ ^h n/'/fes (7/uf fpoi^rs. 



9S 

5 -ff, per sec. Work oi^f 
formc^/c7s for fi^e x a/7a^ 
ilhe u of p, pp/hf an 

the or/'^fn. //M F]r //or A -|-J->— i-^^ 
zonta/ onct ^'(frt/ca/ speeds' o/?a^ accet- 
erat/'ons at the enct ot i^ j^econcts, 
A/7S. speect aoce/erot/on 

f'erf/'ca/ ^jh £p.s.np/7t Ts^p-s.p.s. /eft 
Horhon/al ^^Aps. o/om \ol^ f.p.s.p.s. c/oyv/7^ 

l-^O y)//?e/7 ^as h/om out of cr c/^sec/ 
yesse/ //p/p (? jracui//77y /t ff/oyirs ate/ 
rc?te prcpor/zd/ja/ /c the. a/?70i//7f re- 
ma /nin^, /f th/s rafe n/as iO arams p.s. 
w/?e/7 loo gm. renj^/r?€e/^ /7fiyv Ua^ /?f/^ne 
on/y 50 ^/Tps. remcf/h / 

/J/7S. 6*^3 sec. 

I'-^l //l/t^e/j a /?0X 7S pc^/Zea/ a/^/?^ t/^e 
f/oor bj/ a s/r/Wy t/je JfTla^? 

res/sta/7ce /s //7e prt?afc/ct^.mi^j^t.... 
of /f7e a>eff/c/eaf of fr/ctfon (//7 f/^is 
case, say%)y f>9 tf?e c//ffere/7cr f^/fvife/f 
t/ie ywe/j^lyf or /t/e f>^x aA7</ the rer- 
fie of cor77pOf7er7t of /f?e pc/// or? /he 



<?1 

sTrin^. Th/s /s jc/jf bcr/ancec/ he/ the 
/ipnz^ cc>rr?po/7ent of f/?e dM For 
i/vhaf G/i^/e^ ^ /s Ws pt/// a mini- 
/vu/vF 4/7 s. 4hoi^r 3Z^''. 

I-^^ When c7 s/oz-^e /s y/h/'r^e^ J?y 
a s'fri/?^ fjhat /y/'/7^s (//? ^/7 one^s "^ 
f/hfer^ /Jhf speeo/ co/nponent' 
r?or/?7a/ fd we s/t/'/f^ remaz/fs 6' 
const(7/7f af '^0 ft perj'ec. • The 
iffr/W9, or/'^/ha//^ 8 ft, /on^, j^hortens 
■^ ff, per rfi^otut/on, f/ot/r for?^ dots it 
taAe to w/'/i^ cjpP fFe^araf f//7^er ^j* a 
sT^f/'pr7C?ry po/nt: /4/7S, /?^. 7 sec. 

1-1^3 Fincf the po/or coorc^/nate efcza- 
t/bn of tfpe pat/? of the stone cte- 
sen be of /n I- Z^. 

1-^4 the rope ABCB ru/is at^er pf^l- 
/eys ot 3 orW C o/icf ts maofe Bft:....., c 
fastto the /v^st at B. /f ' """■*" 
the rope /s yvoa/7ct t/7 at A, 
3-^ ft per sec., honf ^t /s me\ 
e/7e/ of tt7e hoo/77^ C, rising i/vhe/? BC 
/s terett ^/^^^ AB^30ft^ one/ >4c 
"^35 tt Ans. \^ fn. per sec. 




\00 

I- Si 5 Dec/t/ce. o form t/ /a fpr centrf^ 
peta/ -occeferat/pr? /h fh/'s rvanner'. 
Form express/ ens for jfhe 

X ^/7//, >^^ ^ of a po/rrt 
re^oly//?f aboc^t /he or/p/h 
/h a c/ro/e of r^of/i/s, f? 
CfT?^ m/'/fh a aonsfff/jf a/7^c//ar 
j'fieei^ of CO (o/ne^) rc7^/'c^r7S per j-ec. 
hho^ the horizon fa/ ar?^ verh'ca/ ^c- 
ce^fero^'ons, Shoyy fh/^f fheir resu/f 
t?nf po/hts fovitar^ fhe or/p/h an(^ 
fhaf its mapn/fc/a^e /s RoS^ ap.s.p,s. 

1-^6 Locafe the fp/p/Kest po/ht of 
the Cahd)o\o, i^hose /^\,B 

e^c/af7o>rp /n po/ar L .:J..^^.^ 

coor^/rjo/es /'s ttrfs : \^ ^ 

A/7S r^3a, d=^liO'' 

1-^7 /t'^ttJhs. /s the tn/ork c/one w 
compress/hp a. ^as^ (^aJ i/- cuft /s 
jTs trofor77e whe/i i^nafer p /6x. persf, ft. 
.pressure^ tfien cLW^ p^ot^- 6c^t he/ 
ooyLE's L>Aw p>^v /s constant, /f/7^ 
W /h terms of ir -^ atso f/hct W 
/n terms of p. 



101 

1-^8 //^^ near Gf^es the coT/e 

y/X -I- ij =■ yfa If 

Oi^^e f£> the or/^//7? /[aja . \^ 



i-^7 The ra/7$;e of a proj'ecf/'/e /s 
\6l^'*^s/j7^(9 /r7//es /'/^ ..-'-—♦'.. 

w is fhe speec/ af /he X6 
rnf/zzfe /h m/, per ^ec. \^ljgn9q_^ 
Hoyv far Gt^rf ^ ffur? rho^f i/v7t77'S 
mi/Z7./e speecf of -^ooof/lpiS'.}^ \^.6Sm/\ 

\rZO (A lens prMem) C, F, L cjre f/')(ed 

po/hts. Cf- 5 c/77. CL- ^ L| -^ 

e/7/, I /r 50 c/77. from L aryd , .;/'' ^ 

Qpproac/7wp if at fhe rale © f-'^^'p 
of SKcm.p.scc. At \nrhat ^""^P 
speed /s O /?7oi//f?p a/7^ if? yvhot 
c/ji^ecf/^W? Ans. To the /eft -it cm, 
per sec. anc/ c/cn/z? i^ cm. per sec. 

1-31 //pyr /fi^h^ a /if/tt, L^ a^/i/'es the 
best i//tyr7jina//i>/7 fpr appaJT? %}- 
o round Of 100 ft c/rcu/ar ^rass ^!\%- 
p/it if the i//ur77/r7at/c>/? ts tf\ 
d/fect/cr^ as, sin^-^r*'^ ot (V* 
ap£/ po/ht /h t/7e path^ 

•^ /fr7S. J?^35ft. 4i/7. 



Definite Integrals 

The /hte^raf , fdFCx) s FOcy-^A, /> 
ccf//ec^ arr IwoepiNiTE ihfe^ra/ on ac- 
cot/nf of fl7e c^rrknoi/v/? ni/mhefr /\ 
aa^oM' as a consto/r/'i?/ /htB'^ra/tJh/?, 

A spe'c/a/ a^a^ ^^Cy <^orTi/'e/7/fnf' pray 
of ofeterm/hf/if ffie ya/c/e o/ th€ 
consfa/rf i/v/7/ /royy he a/e^efopec^. 
G/Ver? (p(^) ol^ =: /(i?c) ol-x. aai^ 

Subtract/hy i(l5)'^0>)^Ffic)' F(a) 
Th/'s result /Tfai/ be put //? the form 

or in fjj/^ /vore compact form: 



103 
thj's Notation bem(^ adoptee/ 

7??/s nety sc//7jhc/: 

J" tCu)du 

/s Qa//ec/ the Definite mte(^ra/ of 
■fCu) c/u. From P to Q. 

f(u)ciu /s fhe Integrand 

U ** *' Var. of iNTEGI?>\T|OM 

P " •• Lower Limit 

CL *' '• UppEf? LiKJ/T 

Note: /he Of^DER m i^hwh //je c^per. 
at/b/7s /hc//catecf fy/ fihe Der Int, 

ff fCoi) du 

must Be per/ormec/ /'s t/?js: 

I. /ntepra/e fCu) du 

JR. Subs/'/ Me the upper //r77//' 

3. Juhj'f/fut'e the toi/i/'er ///v/t' 

-f. Sub tract ta^er n^su/f fro/r? /irrver 

rtie oonsfant of /ht^^raf/hrr, yy/haf- 
$i^er /'ts ^^(7/ue , ea/7ce/s puf u) 
eyatt/afih^ ^ Definite /hte^pz^t 



104 

rhe f^esc^/t at fhe jl^tpfUn? of pa^e lo^ 
may noyy be sfafeaf fhc/s', 

Inte ^tcff//?^ Sof/r n7fmi?frs of a/? e^c/a- 
T/o/? i>y Dbfinite. fr7tefrafs, lA/fypsr corrv- 
spona/h^ f//7jfts are C0RffESP0NDiN6 val- 
ues of ff/e i/'ar/aS/es of /nfe^r^f/o/7, fs 
efcy/^fafe/yt fc> /fif^^raf//?^ ^^i?/ cfe^r- 
/77f/7//7^ 'f/7€ consfanf of //?teyraf/6/7. 

In the s/oec/a'/ case i^fp/c/^ occurs /7?o3t 



ffefc///^ecf y /r Yerrr/s of ^, 

Tf?c correspo/7cf//7f Uppe.f^ ///77/fs are fhe 
Sen Ef?AL Valves, y a/7^ It ^ i/irh/c/^ f/if 
fe^u/re^^ forr7?c^fa fs to 00/7 fa//?. 

Herrc-e fi dcf ^ \^f(^)dx 
or y==- ^ ■^J^'-PfitJ^X 

NotE thaf fhe 'Z /r? fhe c^jipp erf ///?// is 
(7 i/'afc/s fp i^ sohsfffofecf for th-t 
-pC /h ff/e /hfe^raz/af^ yyf/zc^ ftf/^r /s ffyc 



105 
van'aii/^ of /hfe9ra/roh. The /after f/oes 
nc^t a/'j^ecrr/r? we n^scy/t, ar?/^ the efcja- 
f/'(?/7 m/'^hf e^^a/^ i/ve// 6e i/vr/T/ffr? 

Th€ i/a/ue ^7^ a Oef/nif^ Inte^rc^f 
^epe/7c/s ^/y/y (yppn the L/mits crn^^ 
the FoHM (^/ fh-e Iktegr^nd. 

^xa/77/p/e: Tl^e speec/ of cr /ra/h Af RO 
mAp.hK c?t ^,/OF.H; it /'s /hcreas/h^ 
at the rc^te -p^ p7/,p. hr. at m /n/h. 
past ^. Ref, : General /or/vu/a for speed 
So/c/t/o/yt (y^e ^A for Speed /h tv/.p.hn 

m f0r p?//7i/tes past k FM. 

dm ^5 2.5 

Inte^rat?/?^ f^etJv^e/r aorresp^nd/hf ///n/tt 

fhe HALF-BR>ACKErS with th^ LiMJTS 

^/io(/l(d (7e c^seaf fo /hd/cate that tt?-^ 
fast 3 steps anf st/// to t?e perfpr/7ie(;( 

FihaJt(/\ ir^ 06 4- ^) M/.per hr 



106 ^ 

xJ"! Jo Ci^sp^dx = 1 

^^cT c7/7<f yv/perr X ^as one. Am. 5% 




^"^ n ^^^l\^in^d9; solve. A/JS.if^tz 

/he yrh/e f3m beTiA/(Sfn 630 P.n. and %3o 
P.M. if /he ra/c a/' y/Zp/c/; /f/'s ii/rnec/,^ 
cu ff.p.m/?^ /s^ (7f t m/>?^ a/fer 5P.n., given 
bi/ we forp7i//a (X-\Qo)'*'~ 60(530 -f^). * 
Ans. ^5,000. cu. ft. 



Areas 



107 



/? c^r/a//7 fype of A^ea /s of great jm- 
portance on account af ff^ c/se //7 the 
represe/ifaf/O/? of /:f7i/sica/ ^uantif/'es. 

Speaffinj of fjhe xy-p/ane of Anaf^t/ca/ 

map J the f(/pe of (^rea filter reef to is 

Boc/nc/ed 0^7 the 
WOfMW l?(/ ff?€ cutre, u-^fm 
SOUTH b(^ tfie x-ax/s 
^^^,^^^,^^^^^ WEST b(/ (7 ffrre, x= a 

y ■ 

T(p /ofve ffre proi>/em of cafcc/faff/i^ 
^/s crrea, co/7s/cfer f/fst tfw more 
genera f probferr? of Pr? area ff/re the 
abo/€ bc/t t?pc^/7cfteaf a/? t/?e ^A%Tt>y 
a l/^F?MBLE yert/ca/ f//7e kvhcse x- 
Qoprof/pcfte^ yvit/ be oaffeaf x ft cf/s^- 
t//?yc/fsb it frm the t^ ^secf as a 
rc/rpn/r?^ co ep ret f note for tht i^-ar/cc^s 
po/hts afor?^ t^e c(/n/e y^ ffx), De/jote 
fh/s Hariable AnEA^h/ A, or?£t sif?ce 
it ctepe^ais upon ^ ki^e mai^ piyf 

A^ F Ok) 



108 

Now (pfefermihe the nff^ at which /\ //7- 
creases as the eastern ix^c/fjf/arcf /s 
shf'ftec/ cf/onf\ //jat /s t/h^ '*'%^ 

\y There is app/pfj whcse 
. x-coon^/hate w/// Be 
j co^/ec^ X^^ on fhe part 
^^\^ of the curi/e pi^erAA 
-zr^ fhrPt/^h yyh/'ch a /ei^ef^ 
"^ ^ hhe ccfrj be c/ranrn fhat 
yvi// sauAf?E opfr A/\ /nto a recta n^/e 
e<fi//va/erjt /p A A /h Area, /ts^o/fy- 
fc^cte /s ^C^^) y /is width is A%, 

iVoky iviioyv oi/t fhe rer77a//7/h^ steps 
(sec p. \ 8) fer fin^i/i^ ^ ctiPferent/i:^/: 
(fte^dj AA=- -fC^ )-^5? 

Hfie /lak'eA^O wiien ^-a. tfe^i/iredM? \0thef7 
^^h. tnte^ratfi?^ hefi/i/een /imits'. 

To cLA - li H^d^ 



on M;^l^ ^'^^^ une^ertfmf(A^\ (r.^\^^ 



frorrr ihe equotion ^f /H i/i?per houm/aiyi 

f'/:(?rr7/7/e I F/hai the area hetkveen fi^e 
Ci/rve\ y rr ^ Q-x^-ya/?^ the^ ^ - cfK/s. 

Sofc/Ti'or7> 
P/ot the cc/rye roi/^/t- 

re^uirec/ area. On 7I 

the c//afra/?7 a/yvays /h^jca/e the ^er- 
t/ca/ OJ7/?/ /i^r/zonfl^/ sca/es use^ ar?(P^' 
c/rUiV Q l/n/Y sfc^are (Glottec^J "fy shoHr 
ffjc AHBA'ONiT //? ferms of yvhich 
7^€ /Ifx^//" is yp /be eKpress<^a/. 
The easf a/7a^ inresf Jboc/nc^/^ne^s ^re /h 
lt/7/s case mere po/hts, ^-C, or7afx^l 
The shaateat area /s 

tf the UNIT squAFE /s of co/7yef7/enf s/zc 





110 

shp^ the Af<BA^scALE by jf^c/Zcatm^ 
/he (7rea (?f a ct^rik^en/enf' recfarj^Je. 

/f fije C(/n^e ^^/^^ Jbe/oyr the ^- 
ox/s fhe //77efra/rudjc 

\ a, -fa / •'a*' . L//e/c^s the /7^ 

atiVe ^/ the aofc>cf/ 
areai f^r, on pa^e 
i06, /h the proc^ 
fhe (?/fIti/ate wWjjof de fi5c)^ which is 
nefiaflye, 6c/ f "fC^) ^ which /'s positiire, 

fxc?/vp/e : F//7^ the acti/a/ are^ 
hoc^/?ctect /b^ fhe ct/rve iO'^(giK-a)'(x-'M)(3(kf^, 
the ic-ax/s-, cr/^a^ the i/ert/ha/s at yc^a 

a^c/ ot'x- 3 a 
So/i/t/&n% 

"^ ^ A I'^Z^ ♦IS ax- 15 a*-/^Oz.cc-^) J 
etc.^ etc.j (?/7^ f/na//y 
A,-f>A^= .OS^d -^.OQAct^ .oho c^ 





Iff 

K-1 Ffhc/ tii€ ar^a c^/7^er the pa^a^c/a, 

ncaf of x^\%p. Ans 7X^s^.c^fy/s 

K"^ /7W /^f area of one arch of 

Ans. g sf. un/fs 

H"3 fihof ihe acft/a/ area befneer? 
tfie^para^i?/a,y-^'^-SX'^\7^, the ;t- 
oxif, a/7at k'erf/cff/s at ^= / c^nc^ oc- 9 
Ar7S. ^i + fO|^+^7=-40 s<^. c^n/fs. 

K"4 /y/7y 7%f arm befyyee/t ^e cc/n^e 
11=^7.^^ the Lf-aje/s, o/7a^ fi^e /9fir/zcn- 
Ta/s at u^^ and Tf^4. 

K~5 Fjhcf 7^ c7recf uncfer the efu/ /af- 
ar o'/ h^petb^/a, Xj7 ~ i^ 
fr(?/77 fne y^fr//ca/ af * 
^=i 7b ffye yt^rf/'c^/ ^ 
yy/?^se coi?/z//nafe /s x, — I 

K~"6 f?na^ /j^e acftyal ar/^a ,fnc/os'Scf 




bcf the curiae, ^^=-*^-^ C-5c*--2.), th^ %-a^is_, 
Of net the I the -ic^/, 

Ans. 34-if Sf. units, 

f^-7 Fwd the a re a under the curv^j 

frorh ^=0 lo -Tc^cL. Ans. • z'e °^ O'' /•/?<** 

i\'d Find th^ Qreci jnc/iJtded hehtveen 
th€ paralDolQ^ whos-e e^c4 0itfon& otre, 
u*= 8:3t on a 7t:*«Sy. 

Ans, Zl-k s^. units. 

^"^ ^Jf difterentf'at/njf J verify this formula: 
J/oSx* d% »=^ -f /a*-%«- -i- f^arc sin ^ 

Ttfen find Me aroa of 
the qudther of- the' 
cmciE; •%*-*- w^ = ou^ 
whith lies ih th-e ti'rst 
g u ad rant 

K-IO Fmd the Qreoi between the pa- 
rabo/a, :<^= <^-%--3}* //?€ 
-x-CfXiSj and the fine x^pf. 
Answers, 35i, or ^^ic. 





KHl Find the Qre& be-hv^een the 



two 



V?L H- -iTf « Vo. 3 ohd ttie 
coo r dm ate axes. 



1/3 

curye^ 



L 



Ihen oalcuiQte the 
ateo pf one arch 
of t/j e ocfc/o/'d whose 
parameter equations 
Qre: f x = a (e- sin 9) 

/ u = a O ^ cos €») 

Ans, Sna'- 
NoT£^ The area of Me G£Ne.RATir<G cmciE-%o^ 




e^uQ- 




H-13 Get a pair af parameter 
tiotjs f-or a circle , ^ express- 
ing -x (3/?at ^ fn terms of 9. 
Usin^ these, work out the 
area of the circle. 



K-/4 Find the area betiYeen the curve 
•x^u-log-y^y the -^-axis, and the litiei 
-x-/ £7/70^ -^«e -^.7/ a... Ans. One vNm 

\i-ib Show I'f^^t the entire area oF the 
curve , ^*^ ;«?•(■ a^-xV /'s ^ a? units. 



114 

lNT£FiPffeTATION OF SLOPES AND AREAS. 

The Problem Sets £), G, and 1 hare su^- 
qestisd hoi^ yofrious ore the coises ih 
which phifsieoij scjence /3 conc^erned 
with quQh titter whose derivqtives or 
infe^ra/s ha ye some imporhant pntjs- 
tea/ meaning. 

line- second method noted on poc^e 3 /5 
often the most prcnetica/oJe wolj oF re- 
presenting ^ funchfon defined bjj phjjsic- 
a I phenomena, In represenh/'nj a func- 
ttan Qfrvipt?iea/Jcf, the hor)2onta1 coordinate 
is used for the time ot for Me kariahle 

UNDER CONTROL- the INOEPENO^NT yorf'ab^e; 
the dependent yarfafole^ or function'^ 
plotted as the yerhcal coardmatt. On 
the axes of such a dja^ram ft /,s 
necessarcf ho note on eqch one not 
oniy the numerical scale bat also the 
UNITS ernplocfed. 

Such a diaqrain represents df rectify 
the valuer or the td net far); and in 
manjj cases the suorb of the, qjaph 
and the af^ba under it represent 
related phytic qI ^ quantiti'es. 




115 

for example: on a Force -Distance di'o- 

<jram, the ^raph shows 

direct/ (^ hoi^ the fore e j/b.- 

varies as dfstohce /s 

pcissed oren A/O^V in Qhj^ 

CQse where F remoiihs constant as D 

varfeSj the force- distance ^orhrelaiSon h 

F ,c /a D - AW ^-^-P K 

If the force varies 

use ffje re J af J on ^'^•' 

f A vera^epyA D ^ A W — r -^f^ 

from which follows the difference qc/o^lent 

Then make the ftiteryofl AD 
srna/Jer so that 40=0 gnd _ 
we have Ava. Fgrce. ^ Bxact 
the point toward 



! :F 



;. Ft. ibsj 
— f — '■ — '-^-^ 




^-Ahoi^^'^-n- i 



FORCB at 
which AO is shrm/fin^. 



Aw 






H^e ha^e then these two /mport'Oint 
exact relations in case of rarfabie force 

Hence: on a wor^^-distancb diaqf^AM 

FORCE 'is represented by slope 

and on a force -distancb diagram 
woRH IS represented bcf area. 



\n oetiorQl a re I a Hon of- the ttfpe 

H ^ A a -T AT^ 
which /s ex etc/- i/i/hen R fs con- 
^tcf nt J leads, bj^ reasoning h'He Ihe 
precediri'cjy im case of van able H, 
fo these two e.kact equat ions\ 



R =• 



d>T 



Since manLj fti notions must joe used 
for yvhich no fornnuloi corn he found, 
these relations are of ^reat vai- 
ue in that tjneof gfve. graphical 
meQns (the measurement of slopes and 
areas) of find ma deri^atii/es or //?- 
te^rals of onLf function whose ^raph 
can be put on paper. 



Slot=>e: tncjif be mectsu red 
by drawing a tangent 
and dii^idin^ the rise of 
a segment bcf its run^ 

ARBA moLj be measured 
bif means of a plant meter, 
it) J/ counting "^sgu ares'] b^j 
Sirnpson'^s Rule, and fojj 
various other rules. 



graphlcalh 
s 




^raphicallij 




m 

K~lfc Tt)e force necessarcf to compress a 
Sprinq IS zC^ -<) /bs.p •% fnc/7es bein^ 
its len^l-hj ycrn'oib/e during compression. 
Mo/re cf ^of?ce- distance d(AGram and 
find the ^o/fff DONE ^rap/?/ca//(f, 

Ans. J4 inch lbs. 

It-I7 Dran^j as OfCcurcfteii/ osyau cqn /^ree-hand, 
Me HEiGHT-TiME DtAQRAM of Q hctfted ball, 
knomncf tliatit^ horizomhii mat/ on is unitorm. 
From I"}} is dm^n^m find the verf-ical companent 
of //5 speed at a point hoilt-Vfacf up. In- 
diccite all dah nec&ssarcf and et/t unfta 
used caretui/tf on f'i^e dia^roim. 

K-/5 lhf<^rpret siope on a dfa^^rarri 
where u represents worh don£ and 
X represents hme, Qi\re th^ reasoninQ. 

K-H Interpret A f^e A on a d/oQr&m 
where %. represents t/mp and i^ 
represents f^atb ok flow, Qive reasoning. 

h-20 On or pf?£SSU/fB-iroLUME D/AGR/^M^ 

A HE A r&pre s ents \^orh , 3 oj^le 's I cth^ 
js f*xV» Const, Show f hot i he wopk<- 
yoiUMB formic 1 involves Joq^ V, 



U8 

Scalar Slopes and Arte as 

Majntf of^ the (^uanhtfes used in PhCfsi- 
cqI sc tehee are Scalahs^ that t's 

c^UQtif'ities which can t>€ represented 
afon^ pf Twa-WAr scale IiHc that of a 
thermometer; a certain value beih^ 
de&i^nateoi qs tJie zeno^ the others, 
ore des/^nated as -+• or — accard- 
injf to the v^atf m i^hich thecf differ 
fronj t/7e zero, T/Me, Tbmp£rat(//=^e^ and 
Work furnizhso (the negative is Wdt if con- 
sunned), Slop&^ Altitude j,.,^ are Scalabs. 
Vectors^ I lite distance^ hrce, speedy ac~ 
celeration,,,,,, moij/ be cons id ere a as 
hcaiars when only On£ dif^ection and 
its oppostTB. are to loe constofered. 
Some ^ucfnt/'t/'es, density^ ^pecittc 
^rat^itcf, absoiute temp.,,..,, are pure 
numbers, be) n^ never assoc/otted with 
Q direction or with a •*- or- si*^n, 

Slop^ is 6cAi-ARf bein^ defined as 



for' Aij mau be — when A-ic 
is + Qnd so make -^ — . 




#HnF 



IIS 
Ar^a is Scalar J hefn^ defined as 

since it is — 'if the ^raph J~WMW^ 
IS beloyv the axis from a fob ' "^hIUII^ 

3lore and Area are, then, so defin- 
ed as I'D \/r<^/i represent phjjsfcal scolars 

A roRCE-DisTANCB DIAGRAM wifh a ^rapH 
dipping below results 
from a case fn which 
the worffing hrc& is 
overcome and reversed be/ the re- 
sistances^ or springs, So thai" work is 
be/n^ UT^DONE. from h to c. The total 
work that is done and stays done rs 
Actual AR£Aj atoio~ Actual Area, htoc 

J f, ci% -j^he^Qh'ye cxi J, F. d'^J 

=f>=r j_j 1+r /_/ ],r j_ r ] 

Hence the Definite inte^rol from octo o 
takes proper account of si^ns and 
^ives appropriate meaning to scalar oirea. 



no 



Account- for all negative result s^ fn fhis seA 



hrc es 

?•« . .A 






K~-2^ 5-FM Is Qn equation between 

measured tm pounds, ycis in fK /^ 

5x= 5000, A-^oo. Calculate 

the m^rk dane Jojf Jr ih reducm^ 

% from I tt. to b inches, Ans, - 30 ts, ttlhs. 

W'ZZ Atfde-mill pond, c0/itcff/i$ zoo,ooq,cu, 
ft. of water at noon. A ^ate Is then 
opened acf/f?/'th'nq the sest at the rate 
ot []ooo cos (^-^ ^}r'adjca. f•^. per 
flour, f» bei^Q the numtet oF hours 
after noon, Ho^ tail ts the pond at 
cf P.M.? Ans, /(fj 000* cu. ft, 

f("23 The force necessarc^ to thrust 
a ^Q fh pile down into the 
wqter is (70 ^"^ 3.Z ) tons. 
How wuch work is required 

far jusf' suhmer^in^ tti& en- 
tire pjle^ 



Ans "iethtons. 



' 










• • 



H-24 Interpret Af^Efi^ on 01 DeNSny- 

VOLVMB D^AQRAM, 

y^-ZS The upper er7d af a ion^ sprtn^ is 




woi^ked hif hand so as to q fve & 
attatched h hhe loiter end a v'er- 
tfcQl a cceje ration, tvhich yarTes 
^o as ho be C-5Z, -f- ^o sin 3n^) 
in, p. seep. sec. upy/ofrd oh //?e ^ 
end of I seconds, Al 'Ihe. 
end of one second fs fhe 
ho// above oy be/ow ifs sl-a^hna point 
and liow far? Ans. 32 rn. be/ow. 

K~-26 Interpre/- slope on a space- 

TiM£ Of A Off AM. 

K-^7 Interpret slope and ab^a 
on a speso-TiMe o/aqram. 

H -Z8 Interpre/- a negative Arba 
an an AcceLERATj oN--riMt diagram. 

H-Zf Interpret Siopm and 
Oh ft^is d'i a ^rayn J wfjere >fs 
t/re tiorfT-onfa/ represents 
Q temperafare and //^e 
c'drresponct-rnj} yertrfcai/ a speeihc heat 

W-dOFfnd tfie value of \[{S^ di% 
tcf the he/p of- a diagram, 



AH0A 




ttz 



Thb law Of' THE Mean 




The statement' /s made, on paqe 

that thete is q point 

on the curve Pa thrffUfh 

which c^oin be dtaivn 

a tioirfzontcil tihe tbif a mrv 

which the area TQiloo^ is Squared off 

into an ^Qi/Jt^ALBNT rectain<^/e. 

The Qltftude of thi^ point nnvltiptted 
bjj the base, ah, ^ iVes >he areoi oF the. 
figure Paab. This, alhtudeis. there h re. 
called 'th€ overaj^e or Mean Qltitud-t of 
the curve from TteQ The statement re- 
ferred to above maj^ now foe pat thto 
this form: 

One of the values of an altitude 
as it varies continuousftj in an 
inter vat ii the mean i/atuc of the 
altitude for that tnterval. 



A^afn: if q continuous 
from y^ to Q without 
sharp corners^ there 
is Q pornt on the seg- 
ment 'P$5 at which 
a f(^t7fent can be. 



curve runs 




a m' 



droiwn thah is paraUel f-p the chord PQ.. 
Vie. slope of^ this chord /e the hotaf n^ 
rise, ^rom T* to Q, divided buj the run, and 
fis 'slope rs the ayerage or Mean slope of 
the cur^e trom V to Q. The secona state- 
tmerit mc^jj nei^ be put mto ttii^ form: 
One oF the values of a i^lope. as 
it varies contfnuausljj tn an /hter- 
vat IS the mean value of the slop^ 
for that fntervaL 

These two statements are AxibMArie.Thecf 
are summed up tn this single ^ general 
AxiQM^ ca/Zecpf the Law of thb tiEAU: 

One of Hie values of a FuNCjion as 
it varies contjnuousli^ in an inter- 
val is the t^EAN value of //?e func- 
tion for that interval. 

Regarding the diagrams a€ graphs of 
the function fcxl leads to thsse two 
forms of the low of the mean; 

Derivative f^m: iChyfm ^ fb-a)f{m')/* ^*'*'* 



m 

Taylor's Expansion 

Ta/re ancj two numbers^ and their •Sum: 

s ♦ p =■ a 
Girxjd conHnuocfS Sanction ic-ic)j and /Is detiv- 
ahVes, fho^ fi%); kyri^e do\^n thU ec^uahotf lYif^ 
f(s)+pf(s) '^'iF'f'U) -*■ ikP^R - fcs)= o 
of7d consid&r it sojyed hr f? in terms ot>s,p,6. 

Using thiz value of Bj the funcHo^s too,f'c%xfmj 
ahd the nuiv)^r g, form this neiv funchon: 

7^ (G-xfR - i(G). 
The e<^t4(3hdn a/baye. sho^s that' when ic has 
the value ;^=s, this new fun^fmm equol 
to zero. It is zero when -x^-q oitso. Off' 
ferenhahnq the new funchhn with respec/- to x; 

By th-e Law of the. tie an , we mau write 

q)(G)~ cp(S)^ P(p'(m) , Gk!>^<s 
Substitutfhg v&lues just o Iota tried for tfiese: 

Solve this hr ci new form of the yalue of Ff", 



\t5 

Subshtute this in the ori^inctl ec^U cation ^ 
transpose the hst terw, otnct obtQ/n: 

fCG)^ Us-) ^ pf'Cs) ^ i. y^iU ^± P^f'Ym;. 

which is fhe Tayior's F^r^uia FOF^£xF>AAfs/w. 
G /s cf/7jif GIVEN number^ ^ U a v umber to bt 
BUSSTiiUTEO into fCx.) Qnd its deni^/yes;p(s:<i^s) 
/s the number whoie powehz a(:>peQr in the 
expQ/?sio/7j the yaiue of rn is unkn&t^h except 
thQt it 1"$ between^ and s. M is ccil/ed tfye in- 
TtRMeoiATB ABGUMENTyand th-e term m yyhich 
it otppears i^ cq/Jed the FteMAiNDBRTtnt^. 

This expansion is useful in cases where 
1 The R^MAiNOtn is comparatively small, 
JL The ReMAiNOER can be closeljf computed. 
Case I arises if OifS, and fC) are such that 

— F>s G^Sj IS small 

— the tactor containing (m)it small 

— the factorial di trie or reduces the. 

Sfz e of the remainder sufficiently. 
Case II arises when fhe factor containing Cn^) 
has values near erj&u^h oliKe. 
whether its argument /« G,or 5 or 
any number betiyeen c and s. 



T/iYLaR*s BxPANsioN macf be extended to anij 
d&sired number of terms: 

The proof folloy^s the soiine. course as wh&i 
the Remainder k written eiFter three terms. 

The sf^n =^ is used to indicate an 

APPROj(/MATE EQUAU'TY 

In corses i and ir, pa^eizSj vre rrrqjf trrfte: 

^ompJe: Bxpand too lO-h in powers of -k 

Solution: 

We hare aslOi ^ Fsi ^ ssq-p=:/o 

Pt// (j> (%) 55 loQ X , Then cfCs) -= I. 

q?'rx} -^^^lo^e^ (see pajesdSd^d?) 
= .-^3^294*.%'' Get cp'Cs) 

<?>"W -,454Zil4...(-t^^'Z'^) Get q'"{h) 

Ofall-ics -from 10 to ici, x^to makes <p"(C)^) 
^reatesty^z4-x lO'f tfance the remainder after 
the p3 l~erni does not e%ceed 10'^ Then 
log^ 10^ = }-i;4d4^q4.. {jtb "Boo* Wood '^REt>n.] 
= I •t'.434Zq4... i.OB-^OOlZS-t,000 04>s l.(?2il<| 



)Z1 
3^ (f/? QXom(hQh'on of the remamder the chse- 
n ess of A^e djoptoximat/on can be o^certrtChea. 

Ex(^mphi Ca/cu/Qte sin 31^ fn^ffi the vjjue of 
3in bo; expcrnofw^ to the Z'^^powQr term, 
Solution: h'rst note. ,that the expanshn pro- 
cess m^oli^es differentibihah and durjhq 
tms part of- the ivorl^ use toicf ions. (Seepo^edJ) 
Wehaye 5^%= 5236, P5^«.o/745^ 
Putc^(jC)^sin %, so iheh q>rs) *= ,5 
^'(iQ ^ cos X.J (p'(S} - .d>hbo 

(p"(1C) = -Si-n X, cp"fS) - -. 5 

<p'*i-%)-'-cob-x, and (^"(m,)is between 
-cc?s 30"^/?^ -cos Slj' that fs -Qbho md Sb^y 
Writfyi^out the expcinsjon with remainder: 
sin 3/* = .5" 

ahd thiz rervtaihtiet term 'is hs^ than 
^('Oiq45) jcB^dfO or ,oo ooo oj7 

l-l Expand cos>L^^^f'^"'''^ to the term ih pf 

l-l Calculate log^O,oi) from log^ 1^ us/n^ ex- 
porrston to 6^* pmen Bshhiate th^ remainder, 



L-5 5cpaf7i^ tttnoc"'"^' from the term txnO 
fo the term in ^f 

lr4 ^mp/oy TayUrs fprmu/a fd expand 
(^^f<x/ //? pc^t^<?rj of %, 

L-^ Expanf/ o^-^ot /h powery af^-\). 

L-6 Expand <?''^ /h poKirers of -x. 

L~7 E^Dff/7d /ogf cos ^ , (/s//?^ s^o*, far 
e/70(/p^ to yet m^o /70/7-ircf/7fsk//jp /^r/?73. 

L-8 Shoyv that the diNCMiAL Theorem /is ^ 
speo/'a/ c/fse of lay/or s Eicp^/is/or?^ atei^^/op- 
/n^ (p-^hf in p^ir/ers of b. 

L-9 Ascertain, tjy mecfns ofaTaytofs (e/mn- 
s/0,% yvh€tt;er Oc- sinic) /s more or /ess 
tt7a/7 (^-^6) i/vhe/r -X /s /?ear- Tierv, 

L-/0 Compare (t^^- 1' ) o/?d !^(i'Cos znc) 
when "X- icr*^ iy expc^nct/nf /n pmers ot% 

L-ll ftow accc/raie/y ca/j\/?(d^Z.H6,^)jbe^c^nd 
by me'a^s af ar? expans/on //? por/ers ofi^ 
from e^ to t/7e term fn (if ? 



m 



The rerm sec(o) an^ carrx//^//he ex^ 

<^/f /7Z?/77 The e)ipa/?.j/o/7 o6tff/m^w ir['^? 

L:/4 Jhm fhaf the expa/;s/m ^f ttt 
/^powers of^^roc/forhfomc^f, % 
jcfe/?tw/ yv/fh //rat oi^f-me^ ly a^/y//,^^^ 

L-/5 £xpa/7c/ 6/^7 ppi//ers of^. 
^^^^ f^jPP/y ^e proper exprcss/ons vh /ihe 

L-^O £xpa/7^€ /b r^/?7a//jafer /h ^. 



130 

Tavlors Formula ExTENoeo 

foyUrs /^r/7Ji//a fpr expansion ca/r jl?e eK- 
tended fp -/vr;cf/cfns of serera/ yanab/es.^ 
1/7 p/ace of f^e Ti^la/zhn gs.s-^p if irvi/f 
now ^€ m^fT cot7yen/erTt /^ i/sz fih/s /?o/i//h: 

and the intbrmediats. Argument, 7n,r^^e 
pafe \Z5) w'/ll be rvtiffeni ^^ (i^ etc, 

%>< ^l ^^ ^\M> *- ^^' 

By f/je' L^iY (?f i^e Ijjs'an: , 
i/vhere the erpress/on 

means*." pf/'-ffkr-efiti'afe uoc) mih r^Jpecf /6 
^ -fhen r(^p/ace x Sy ^'J in sAorf V *c^)\ 
The f'C) notof'/on Q/i>es n(?f seri/e // fhere 
ore seiretv/ at^msr?fs {Inobp. Variables) 
OS jt o/^es pot mcf/'cafe \/\^hich o/ie^r^ 
olc/ces fhe /hcremenf (see pays \Q, sTeps ' 
♦ and ^ J f/v/77 wh/cf? fhf der/votfve come^ 
Bmp/oy^ fherefore^ ih/s neyy s*y/vi?sfi 



-^<pC y 



doc - ' ^- 

To mean: ^W/^renf/a/e <ff )^ nyan//n^ "ic as 
t/ie /hdepe/f^e/tfand Only rar/dJ^,'* //" u 
ca//ec/ /he ?ABJiA\. D^ffiKATive y/ith re- 



i3i 

spect U 1C. The 3 (a "rounc/ e^e/fal /ir 
reservec/ for /Mca/Z/i^ parfrh/ c/er/y^atim, 

P^j^.z) /na^Je re^ar^e)i/ as a A^ncf/on 
of X (7/one^n>r purposes of compar/so/i 
w/tih <P(0C,(^^:z.)^ a no/ so i/ire hare; 

^y ^he s^^e Mhof of recfso/7//79 yve hai^^^ 

Comtinln^ these /)y Jv6st/Tu/?o/?^ yve Aa^^t 

l^nf//?y^q:> for cp(^,y,7.) a/?cf A cp for 
prax/maf/on, /rA^,A(^,Az. ar^ jma//\ 

l-Z-i ^(f^-i^)=? L-24 -^\fr^ =f 



\3Z 

ApPfiOXtMATlON JFOHmiAS 

EKarvple\ Ojb/a/n an ap/:^r0xmaton Armt/h 

^r fhe sft/ore or the cosine of ff/j 

an^/e , j'/j th^ ne/'^hbi^rhooc/ 0/ SO'*. 

So/ut/on'. 

Us/h^ rafi/arrs at first' (see p(7pe BT) 

Pot fC^ ^ cos'^C^ rac/Za/rs) 

^ co^ ^ -- SZ. sinf oas§- 
^ -^5 -.8660 € 

we musf pc/f 5 = /rS-riSO , ancf inre /lafu 

^ .SiS —.o\SSS 

L'Z.7 Otfam ^/7 4.F. -for ftie rec/procaJ of 
a /7c/n?l?er //7 t^€ /7e/^hior/)00^ of ^S, 

L-28 If Z= ^ yeto/i AE for I i/vhen 
r is neariy ij c^na/ q /'s neari^ ^QO, 

L-Z^ Ohto/if 0/7 A,?, ^r the area of a A 
//7 yy^/cA a^\^ln,:, h^^in., on^leC^ ^o! 

IrSO l^orA ^c^fa/r A,t for log^jin *x n^fie/; 
the c?r7^fe /s near/(/ o ri^/it a/7^/e. 



f?5 

Small Cof^REcriom 
s/cfe we f^r/77u/a far a (^c^anfy'ty a cer- 

7a//7 set of STANDARD C0/7M'a/?S a/7£/ 

va/c/es. O^y 0f/7er >^et pf i^c9/ues /scon- 
S/c/er€dfo i>e yof ^y aa^c//>zf Correc 
TiONS 7? /^(f sfa/7c/arc/ ya/i^es. I^ken tfre 
correct/o^s cfr^ relaf//e/y s/rfa// /tu /f^t 
coni^mrrt fo ca/cu/ate from /h^ or/^/ha/ 
formula J Si/f ra^^er fri^m (^/j ap/^r^//- 
manor? formt//a C0r?s/sfj/?^ of f/ie sfi^/?^ 
ord i^a/o€ w/tj a/7 ac/a^^a/ f^r/n for 
^ach correof/drt, 

lir exa/77£>Ui V= ^/.0l4iF(l +5fi>-fC^ 
(7 joona-x/peecf -^r/rju/a^ the no/af/o/j i^e/n^, 

yA/?iABL£,(/Wi"nS, quantity; value, 5TAN0AKD COWtl' 

Nme/ers p.sec.^ Speec/ of Soo^nd, V« 33Z.2 

^d(/r?es>/>,sf,c/Jf,,Atni4>sp/?. Press., P-/, 0/2,630. 

I decrees Cq,, lemperoTore, t-0 

C (jrns.p. cu, cm., bensitf ofA/'n c ^.oo /m 3 

T/je partio/ dermt/i/es of V for c?n(/fmpon 
consist/J7f offocfors) art? m^f eas/a ca/- 
co/ofed hy SjoMfry^ op IoqV, d/Werenfihrmg, 
a/7d mn mo/t/p/(^/^^ hy V 



154 

/"Se A.E /or a correcfee/ yali/e afMh (p.\:^ 

Calcu/ote the cc?eff/c/e/?rs,^ ^ ^,^*i/s^ 

\/-h4Y^33;i^-hoooi60^Pf.6oa^t-ja8ooa4c 

\ys\ GimrT-^nif, /vra sex^oncii penMom 
i/nc/er s'lfcf/r^^ari/ cfi/ic/zT/orrs T-^ seo._^ Z^ 
9^.-29 cm.j a- qSOcm^per ssc, per sec. 'Whaf 
correctlom musf he c^c^c/ec/ to V — Z 
/n cas-e of sn?a// correct/ons AZ ff^fii/^7 

PI 
irdU G/\/fn N^^ j^' ^^^ fPrmt/Ia -for 

L-35 Qiven Q^^^ a^c^Q-^^Y^ H4iert ar«^ 
i/=^^^ci/7a z.^6 , Mat correcPo^s /771/sr 

otcorrecfiwi m\ fo%^ -hM Aij, errc^^o^/oz^ 



f35 

A&SdLVTE EHR0B6 

CaJci/Iat/om ha sec/ up0n measure me^ff^ 
ore Ifcfh/f to err(?rs aw/h^ fnfm)ihe 
/ivpossii?///^ of meas{/r//?a w/Wi ahs^hft 
accuracy, fjhe ef-Pecf ora s/na// trr^rma 
measynf/Tifnf upon ^e ca/cc^Me^ rescf//'m^ 
be jt/dfe(^ fn/77 an (fpforpx/matwh ^firmula. 

If <p(%,y,^) /j; the (^uffnt/t^ m ms/7 /i caJ- 
Qu/a/e, bc/f fhe measc^/v/?f&/7/s 0/ %^,z, 
^iye fesu/fb tbo sma// Sc/ the un/rnfii^/? 
amoe^/7/(9 /ix.Atf^Az.^ i^/7at yve rea//c/ 
Cff/7 ccf/cc^ /are /s nc/' <p(%i^, Tj bt^t'cp(^,%-^ 
which /s too sma// /jcf a/? am^^/ot'' 

The ma^n/ft/a/e offhe tRHORS,A%t^^H, 
etc. J cfepe/ic/s" (/pon ihe least co/nt of f/ie 
weasunhp /tisfrumen/}, 7ht/s if x ^80cm 
measurec/ m/i7amsfersfj'ck,A% can hard 
/^ Ae more //?a/? U cm; /ihe J/ir/s/c^s on/he 
protractor i/sec/ -^r & sc/^^es^ t/?e est/- 
mo/c for A^^,,,ff/c^ 77?c aJbai/^ for^i^/a 
asmpst/7e ajbso/t/te efr^rA<p^ to its sepa-- 
rv~^ sources^ ar?if ena^ies us to co/77jparip 
tjf?e e/peats c?f the se/(frc?J ctata - errors. 




\36 

ExGrmp/e: The fiei^ht' of P aboi/e B is 
ca/cu/afec/ fro/77 the farm i// a 

^ ft=: in-- C0S6) 

I js cerfa//?/c/ Cffrrecf fo ImM. 
ond 6 to \\ //on^oiboaf- K 

A ^ ^ AlO-oo%e) -h aO'I Sin G 

We ha\/(f Al<.l;A(9<.0l'(^ Cracf/ans^f) 

.'. Aft < *00B -'-•77 or .77d«"^' 
Ohji^rve that a mc/ch rooq^er meas- 
i/rerTierpt ot 1 /??/}A/' haye Been ^ ma^e^ 
j'/zjce At ci>n'fr/Jb(//es com/^arailye/^ //tte 
/^ f/^e ad'So/t^fe err^r of thensolf^AA. 

L-54 Co/7?jdare t/je e/fed/s t/po/? E - ^^ " 
of errors /// ffte cfata /f y^^ zpoundak 
ir^ ioftpersec.^ a/7^g^5Z. ft'persec./?er 
sec.^ ar)of Ayv<.t ^ Atr<iZ-^ -Ay<*^ 

\j^35 /^otv ser/oc/j/y ^oes a s^/na// erri^r 
/h measc^r/hf t/^e o//a/7fefer of a spAere 
offecf Its cafao/ate^ ro/i/rrre F 

ir36 t>/}cc/ss t^e coMParat/ve effect of 



157 

T-^, the errors 6fm Att^^.ooi ^61^. 1^ 
AT<*0I , irvhift is the exrreme poss/tJe 
error iio 0? A/?s. 11^ 

L-38 /f v)re msh to cte/er/nine ^- TTni 
/h ^ case lArhere T^,0\ cm.^ (9^ 60^^ 
r^,Oicm,^m^ 1.47^ atifif g ^ qeq Cfn.p. sec. 
p, sec,^ yvhi'ch of Wese qc^a^f/^/es »s^opt^ 
oe measc^re/a/ /?7osf cfcci/ra/et^^ 

L-3^ 9iyer7 B^imir": fihd E h(^ th^ 
correcflin /?7€^hoc/ w^/ier? m= \.oo\ ^ra/?^ 
and ir^^, oo\ cm, per sec. 

L"-40 Caki/bte ^ tf m-3me1ers(yYithin 

.01 c/7i.)j anc?t H-USX m. \ 

(mth//7 ,07s c/77.); anp/ ^\ 

compcfre ti?e ePfeaTs 

of tlie errors /h H^m,l,on^, (Optical Le^^er) 

L-41 Show that for two nc/mhers n earty 
^(^ua/, Pond Q^ the geome/p'/c /nea/?^^^^ 
/s nccfrty et^uat ti //?e at/thmef>c 
mea^, ^ t P4- <?j . 0oc/i/e yye/^h//7pj. 



17k J 



\38 

RtLATIVE EHFiORS 

If j^cp /s the obso/ufe errvr th<^, -q^ /s 
fhc /?£MTiVE (or Fff/^cnoN alJ error, anaf 
100 ^% /s 1he PBRctHifK^Ecrf error. The 
forrrtfy'fa fonriect//7^^pie c^hsolutt errors: 
Acp^ ^ Ax 4ifAj ^Haz. 

/J r^/^^//^ co/T/i^rtea info a forma/a 
conrt€cf/hp fhe rt^/a/l'ire ern^rs'. 

iTje con?/?ij/vt/ife ser/oc/smss of /^^reUffi/e 
errors isj'i^c/fed fhm ff^e rrra^/i/fu^e of ' 




/f fiie -ft/rfcf/OP, <p, /s a oroc/c^ct'ofpoi^f' 
ers f/fc cor??/:far/M of r(?e effecn of 
the re/afl'/e errors is reri^ eas/k /nac/e, 
F(fr if qp- ^t^./.z.'^ ... 



X 5^ lo^qP =" a J the exponentpf^ 
T/re rehT/Ve error of eacf? faetor f'sJUe^i 
m(//fi'p//e(^ 6y ihe exp^e/it of that factor. 
Tfie h/f^'pom'r ^c^a/?/?//^ shoM Therefore 
^e measc/r^rf m/J^ e^reateft rekfii^e ifccurao/. 



L'^Z //? ca/ct//af/f?^ S* irjt, t/vhen t^Ssce, 
or?i^ q=^Z.Z ^.p.s.p.s,j yyf// ff/7 ernfir fff 
* I sec> /h t^ or a 1% error ih ^ bemif 

L^i3 //7 fhe slic/e-w/re br/clfe^ (see E Ih, pa^e 
(>\)y x: fi :: a: c-a, H an^ c be/hf 
constants. F/hf/ the percentlf^e o f errpr 
/h ^ c/i/e to a s/rja/t ^rror m a, 

L-44 /n a fan^e/it ^a/ra/70/rfefs'r fhe cor- 
rent /v^ ensured /s proporf/onaJ J 
to ^iiie^/iye/?/ of the any/e o^ ^^---"^pr^ 
f/hd theperce/7fafe t^f error /h Q^ jr^"3 
/he curr^^f e/i^f/^ a j'/natt error /h rer^My :k, 

Xj-Ai 3y /^ea/is of a ^hera/neler the tv- 
ot/iis of a sphere is found >^^t^ 
from these ataia' three /^Tyrs^ 
p£f/n7s on /he surface^ 46 /^^^^-^LSI^^^ 
C,^ ft^rm on eqp//ateral /^, ^i^"""^^ 
i/de^Scn7.;?t^,perpen^/c- 
u/ar to }/s p/ane ar Its cen- 
ter; ^ 4 cnt. Get an erocf forntt/fo for 
the ra^/os, OP, (^m^ an c>pprox/fliahbn fi>r^ 
mc/tcf for /he errors, on^ Joft^e yyh/ch 
^jf the c/at^ sh^c//^ ie most' aecora/^. 



ItrrEGRATION AS 5l;MM/\TlOM 

Bj/ pte DtPiNlTiON S^^^f^ on pa^e 105*. 

f^-feK)d7C^Fa)- fCa) where Y(^^ \Hnc)d% 
ft Mkm fh(7p 

jj K%)d^t \ Jtfef; d% -= U -foe) dx 

fffhe P/4Ne£ (^f i/'a/t/es, ofoh^ he o/mc/ez/op 
fhfo n pc^rfs^ of any s/zcs-^ by /? s/f^tofnofn- 

Us/hy f^e /hte^ra/ Form of fhe Law t>f/he 

or^c/sinf /hf a p/Pf/c^o/' Summation Notation; 

Consider mi^ fhe sum 2, 'f'(Xj)Ax* , 
^>5y64 (^/'ffers frorr? ^e o/her oniy //? 
fh(?fic^^ /hsfea^ af he/h^ fhf cofira/rjale 
of a 5^a4ffJ/VG 0¥? pp/^f as rrij /s^ (see p. 





141 

/f fhc function is re/pn^se/7^c/ ^rop/fica/fy ^ 
f^iTn /h eiff^ifr sc/m ca/r J?e csjorese^f^^ iy:^e 
area of a r(^cta/7f/e ys^arrd/i^ a/? f/^e aorre- 
spondin^ AK^}\^pse upper e^ ^ ^o^/7<^^i^^j^afh 
pf f(%). The s-t/ms 
/ of these s^s of 
recta yi^Jes maybe 

// the mmber of parU /hth pyhicti the- 
rapye^ 5..v.(2, /> c/tt//'c/ei^ is /ticreasec/ m/houf 
I/mit i/7 ^i/ch a i^ay that atf /he at: '//jTer- 
i^a/s appri^aoh zero^ then m// the //injt 
of either sc/m be the ah^a uhoeh the 
cuHVB. As i^as sJtoyy/7 on pap^ /(?9 sc/ch 
an area is erpre^ssijbte as a ^etfnit^ 
tfiteyrat ttejice /he Limit of se^ch a Suti 
as /he sec 0/70^ /pa (/ he catcc//a/^^ /x/ means 
of a otefitjit^ i^/eyr^at Sc^nrbotiTL/^^: 

The use of the J' sip/?, (o /0/7^-Sj /hitia/ 
of the jYor^^SoMj^ tor the s/pa ot /h- 
fef rati 0/7 or/f/ha/e'ot /h the property just 
proveot fo betony /i t^e atef/'r/ite /h- 
teprat. 



t4Z 

Summation Problems 

5(Of7ip/e} fftfof 'fhe ^erfica/ op/npo/re/rf ff 
aflraofi^/? c^r? aZ ^/tz. 3/7^ f of a i^erfica/ 
10 om-y \%0(^m. m'r^, i^hose /o^v^r ^/7ff is 
10 cm. from /he shpf' a^^^ le^^e/yi^/fh it 
Given ffii^ prai^ii^fio/ya/ a-ftracty^ force in 
dc/ries /s 6b*\<i~^xpr(pc^c/ct £?f ii7i^ masses 
/p Qra/rys-T s<^. pi^weit c/isiaA7ce //? om, 
So/uf/on'. Co/js icier /he mre c/iyia^e^ i/jfb 
1 ' shorf ien^ths) ietxc/n.^ tiff 
.-"'v itei^iji' of cfrfy poij^f i/7 fife m'/e. 

%^i B^ r/^e rrjcfs^pf^ cht^rrir is ^ ISO 

ffre'MS; /fs a//rac/fh force o/Tiihe'Si7otis 

65>^ lO^x ^150 X 3 ^WiooT^T 

ai^/?€Sj fije -x jbe/h^ ^e coora^/zjc/ie af 
imf^po/'rt' iife /hferra^i 4^ ibrrara^ 
hri^/cf' if/e aiinr(^^o^ (^rerie/^ iyMee/fc/ziic 
as (? yvircpie /s cZ/r^ec/ec/. 

Hi/ifipiy /^/s ffy_Jjie_reWyjn^ Oipj-im 
%-r sjAOO^X^ 
to yef the c?/T7oc/nt c^^^iri/bc/feai i^ ffri^ 
ciiuni< toh^r^jife fo^/ vtRTxcAv force 
eK^rtec/ u/pon the sir of iry /he wire. 

The/r mafre a ^pt/mf^a^on far a//^^ 
fikscitc^nirs; ^e ^t^i Keri^ca/ c^/rjpo/re/ffis 



143 
CbmpoM fhh by mecf/js of the c/ef//j/te J/ile^rah 

/%^/? ca/ec/fafe /he Hom'z.oHiAi. compo- 
f?efjf erf atfracZ/o// /h the eramp/e (7^0 ^t. 
Ans. 654^ \0^'^ c/y^es. 

W-% ff fhc c/em/fy of a so/i^^if/? /'s 
[^'-rCl-^oOjo^s.perc^.c/r;. at a c/ep/h 
of nc cm.^ Die/oi^ ffje sprface^ f/hcf by a- 
sc/mmcftfd/7 prifcejs fhe rpc7ss (h a jar of 
crosS'Sec//d/7 ts s^.cm., \Ocm. ^eep. 

M -3 . Jb/pe M-1 /f ffje i^ire^s cfe^s/f^ 
\/ar^es cfirectfy mfh /he hei'yfyt, f/rsf 
c ff fee/ faff ^y me pro/Jorfo/raffrQ fac^r. 
Ans. Zj P/77S. Q/?^ cms,\ 6i8x{6'^°afy/7PS, 




M -4 The wet^ht^ m tons per runnii\^ toot^ of a 
maso^rj/ aroh is ^^^.-.^-eo' ->-»- 

% fK frow trie he<f- 
3tot?€, F/hd th-^ ' ^ 

total ^ei^ht of moisonrji.C>imer[s\oins(mfi^ur^ 
Ans, 71 hand ted Tons. 

M'S The woth in ft, lbs, done tn ra/'s/h^ 
the bHcHs for a woli eauals the sum of 
the products of the weight of each 
bn'c/c muftiplfed tn/ ffye distance jtis rciisedj 
units bem^ tbs.mfid ft, If q cubk fod of btf^k 
ivef^hs zoo lbs and each brie H has t^ be rcfk^ 
frorr\ the ground, find Ihe total worH<ion&m 
rof/smj brtck to aqrry a WQll toefK l&n^ 
and 4 ft thick to a height- of 5o ft. 

Ans. lo^ft.ib$, 

H-y Venfijt /sin*0 o(B = -i©-^"5mz0+A 
Then find how tnuch ener^if (- worh danef) 
ts transferred in one second be^fnnin^Ql 
%^0y bcf on orliernatin^ current, for which 
I- 5 .smfzol-^^) amperes, and t-wo, sin 
(zoV°'^)Yolt5^ t beih^ in seconds, Given that 
I amperes at t volts do E't-^T joules of 
>¥ork in Al seconds, Atis, ^'jo joules. 



M-7 The donsjh^ 0/ a r^^cf yar/es from palnf 

io pothtj So i" ha*' at- 01 ddit, * \ > 

X cm. from one end >/ Is " ■^^"'""""""'"" 

^rams p.cu.cm. Caiculate it 3 mass, fts 
length, Ij bein^ 3cm, its crosi- section bei'n^ 
one s<^.cm, Ans, 8^ grants, 

M'8 the weiqht of a beam bein^ Sj^ibs, 
per running wch^ w hoit w 



1 



15 tht moment about ^^ 

the poCht P of the wei^h t m^r-J^ ~"/i^_ 

of the projectm^ jo Peet^ ^"^^ "'^ ""*' 
of such o beam^ one end of vrhich is 
bt4iit f-trrnlij tntc o yyoilL 

An$. Zloo lbs, fh 

M-*/ fiineti'cBneroit^ m qrom-centinieterj is 
efuo/ fo (Mqss in ^rar?7Sjx( Speed in erruper 
sec)^ CoUccdqte tt^e i^metiQ.Qner^i^ oforod^ 
Hoo ^ms. masSf iocm,1on^ whef[ it is 
whiflinq about one srrd at the rate of 
3 revolution s -per sec, ^Sp(\0(^m.tm, 

M-W interpret Si ^f am, phifSicoJ/j^^ )f% 
measures distance, i measures tim% and 
m measures mass. Assign suitable unih^ 



duMMATiON Elements 
In the Qutvmnoih'ah- li'mil' 

ofie ot the single terms m the sum, 
ffjL^'A-K^ is called Q Summation Element. 

\/^heh the desired summation element cqk not 
Joe found eractf if , it is possible, without Qiti(f 
loss of Qccuroicc/ whatever^ to usiz on appro/- 
imahon hr it that c/f'fFers ffvm it hcf high- 
er pon/ers of Irhe incremeht invoil^d, ^x. 
for^ tF f<jcvA>c Is such an oipproxi motion: 

f(x)'Ax^ fCx)*A-x-^ cpC*:).UTt)^ 
and the sumirtat'/on limii- abo^e Qaua/s 

The Qr^unjenton /wye /4r shows that sudi 
a limit is unoilteted bcf suppostn^ oil cf th& 
A'pc's h> be &(^uci/. In that cose the secomd 
summaHbn timih caio be ttanshrm^d f-hus: 

(bjf pQ^e lb) "fl^^l* {Vt"^ ixyA-^ 

f b 



147 
Thh e(^cicihcn is, therefore, cxactly htue; 

In irieitloer oF these summcthhn fmit^ does 
the e^uatitif of tJie A-x/s maH^^^njf ctifter- 
&nce, so tliQt supposition con noiAT bQ drof^p^d. 

Ihe retation beti^een these t^o sum^ is c( 
simple ome to shovK qra phi call y. The functions 
F(x) ortcf tUc) diner onlCf bij f-en^<5 in- 
Voivinq A')^^, oind sawhen Ay^%^o, th^ curve- 
y-rOc) must approach the curve ^^Foc) 
.'■'■• ^— «^ r:^-^% 




As each sum represents the area of- a set 
oF rectangles who3e upper e^ds touch the, 
corresponding curve, the summo/tton-Lmns 
jn each case represent the errea under 
the same curve, the limit-curve^ ^s-poy. 

E^dny/aie; Find the area ot the coirve 
who%e e<^uQtion fn polar coordinates is 

V''l-5 5in i^Bradixns) inches 
Sotutior): 
Plot the curve to a convenient scQte: 



146 



r » ff sirt, Z 6 



To/re as eleiyients f^Qn- 

shcfped pieces. Call fht 

onqk of each p/'ece Zt9. 

Compare Qncfptece yvith 

tl7e mscrWed Qnd the 

Circumscr'tbed secZ-ors. 

fts orea /*s mhermne- 

di'ohe betiveein these sectors^ areas', 

^r-rA^ and ■i:(rfAr),(r^^r)A 




^ ^BsrO 



e 



That the s4 diffef onty btf higher ponders oF 
A © Qppeeirs otl ifion^inj ^r, by the Lqw of 
fhe Mean, h a term in A ^j thus: (pcf^e 13 o) 

AT ^ f(B^Ad)-f(B)'^Ad'i^f(S) 
The rcjuTred orea is, therefore, exocflj^i 



M-M WorH out the ctrea of fhe circle, 
r^(0L sin 6) cm,j b(f q summcfttffn hmit, 

M"/2 Fthd the area bounded bij the h'nes: 
r-secd, 9 =^o, 9= J. Ans, iufhit. 




yi-i3 l/Zork ouf, the /oium^ of a co^e bcj 
mearK <^f <^ >sumnjciHoin ti'mit-j, 

M-H WorH out the vchme of a sphere b^ 
rneans of q summoNot^ l^m/t^ /-''""'n 
usffjj cfs 9/ej7ie/7f-s q set of por- ^^^^^^^ 
oil el slices. [ ^^ ; 

yi-\5 Work out the mass of a sti'ck if !h 
density varies as the sfpgre root of- th^ 
diQhnte from one ond oP the etement coit- 
sidered^ t^e densitif at the hecfvu end be- 
inj Jo fb. per cu, fn,^ and ttre stick t>ein^ 
t ihcf? sQuqre and a (f^rd lono, An$. ^ lbs. 

M-/I? The force with wbfch the sun palls 
on Q i lb. mass e^UQls li^ld^tona divided 
bLj the s<iucire ot /Is ^ /^^~^ 

(distance in niiies tram " T" J 

the sign's centers Fmd — 

th€ work done btf ^olctr attraction in pull- 
fn^ In a. / poan d meteorite from the 
o/t)it of Neptunf, radius MB^lo^ mile^t^tlfe 
surface of the san, radius 43 x W^ wile^. 

Ana. 3500 mile-tarts^ 



150 

Integration By Parts 

Jronnula iz^ P^^ 7^, leads h a Nndcr- 
tnenhn) method oF hom^tormin^ thJ-e^rak 

TZ Jv du, *• V J d^l' - j%c. dv 

/{ycfr/'aJbh factor Represented qb^ye bjf -v) 
may be mailed oaUide th^ S'Sij^n, pro- 
vided Q certain integral is subhracted as 
QH ofh&h This process is colled In- 
TE OR ATI ON BY PARTS, the /at I able re- 
moved f^rom under the S'^r^n, i?^ and 
ftie th/eyrai of fhe remaining fQCtor^ tu, 
beifi^ referred h q& the parts. 

This method is useful vvhen the hew' mte^ral, 

Ti7€ for/nulcr is often writtet,: ivSM-vuL^i-udv, 
/. Example of s/mplet neii/ml-e^raJ: Jlcf^x dx, 

Jloqx dx:= loQX Jdp- J:X d loqyc 
Z. BxQ/npJe of similar new integral: l^^sin x dp. 



151 

- <• 6^C0$ Tt -^ J 6^ Cos -K, ^p 
Apptjf the sante process tv t-he new integral, 
Szcas x3iTc ^ d^Jcos. % d^" J sin % S^dp 

Thi^ neyy)hhe^rcf/ is f-he orl^i'nal one. 6cfb- 
sh'tute into fhe fireJ- et^aaf/mj hari'Spose. 
fhe Se'^^sin X'CLp, oind dfvi'de, bcf ZjOi>kn)iihj', 

A/-/ J arc QXYL 3 <Js W-5 id" sin fLXdp^ 
\\-4 J i^siy? xd-jc N-8 l-^ e^ dpc 



U-^ J'd^'-dx. (used in Hj^)- Use this 
trans fot mat ton I X^op'-x*)'"'^* x^d.7c ^ 
■ - Ua^-x^y^ dye -cd-] id-'xT^"- <lx 

v/ith Uc^-^'^'^T^'^^dp^fa/r/hQ dr^ as du; 
ttans-torm the '[a^-^-ict)'^^x^ wto 
(a^^;*^)'^^ - a* Ca^-^z^") "^^^ ; cance/ the,^ 
ori^ynai in I- e^ rod and solve tor Si^-^TC^T "die. 



152 

FiNOtNG Lists 

n N^B) then 1= '^oSe^i 9^ 

Arc sinCu] » number rad, Cn B if 3in0^Ui ad 

71^ number rad, in ciff^/e-umt; l^^^oifA^^^' 

d sind — ^ cos^A^f ^^ 

tt>rmulois hr Ifite^rahan and diHerentjatm ^o 
Ajfetm/c Differentials^ tcf Tronscendenta/s^ 70 

a=5-»-P, T7^ behyve-en d oiind S, iz-^ JX.^" 
q,iG) - (PCs') -f- F qp'Cs) + -^F^ 9*'CS) •+•*• 



153 
If- H IS the rctfe ohriorease in b per un'tt gf 
thcrecrse w Tj (see pQ^es lo, 33, liB,(ii>,ts^ 
/iSri\TsTHff?/»Te^/?, /7 H/s q constant 
AS-^Al^ Mean RatEj F, iF R is variable 
dSrdT^TRCff^ IfATB, R, ]^ J? Ts EITHeR 



b£FlNiTIONS AND SYMB01S 

Arbi hairy Con st,^ ^ Q ^f In te^ral, 



Arc sin. Arc iarij 68 

Area under, toq 
Ar^atnetitj ^ 

Conh'huous, A 

Critical Ar^,y sz 

Definite Inie^roij los, 

OeriyQhpBj fz 
Difference Quo.j n 

Oiscc/ttirjuitjf, 4 

tlementj lA-la 

^xponenhql, &(■ 

Funcl'ihn, & 

tncremehtj Q 

inflecHorit BO 

(Consult also fht Table, of Conlents)^ 



^8 

hterf7i€<i. Ar^xj izs 
ihyers€ fc/nchon^ 64 
Leqst Goun/f 135 

Lo^Qrifhni, ^0 

N^£^4veAreQ, fio^ //^ 
F<3trt&, 
Parhhil Oeriif.f 

Reryiamder, 

Scalar, 

^hape, 

btandeirdj 

Transcendenhxlj 



%— /\ 



%, 1*0 



da 



/5 



1 



(^5" 



I?- 



IfB 

30 
f33 

J 3d 
)30 



154 



TABLES 



Af^umenUn Hadhtii 


JifATUffAL FUNCTIONS 


Argument- in Decrees 


I 




-f 


+ 


+ 


+ 




X 




K 




A- 


— 


— 


— 




ir 






JET 




— 


— 


+ 


— 




an 






■ 


— 


-+- 


— ■ 


-+- 


m 




SIN 


COS 


TAN 


SBC 





S.I4 


AK 


k28 


0. 


/, 


0. 


1. 


3bO 


ISO 


l&O 


^ 


'^ 


3,05 


3.23 


h.ZO 


.^9 


•^? 


.09 


hoi 


^5s 


IBS If 6 


s 


-n 


Z.<f1 


3.3;; 


ICKII 


•^7 


^a 


.»8 


l.oz 


360 


190 /70 


10 


^^ 


z^ae 


B.4-0 


koi 


^zk> 


^^ 


•27 


t.04 


346 


f^6 


H>6 


IS 


.35 


^ni 


a4^ 


5/^^ 


rB4 


.^4 


.3^ 


Ub 


340 


ZOO 


iko 


zo 


,43 


z.^l 


3.Sd 


5SB 


r4^ 


'V 


^7 


hio 


335 


zos 


\55 


15 


.5Z 


zXz 


a^7 


5qi> 


.5 


SJ 


.58 


Hi, 


33<? 


zio 


ISO 


30 


M 


k.5Z 


3.75 


5i7 


,57 


Jdt 


^() 


hZZ 


325 


zi6 


U5 


35 


.JO 


Z44 


3S0 


5.5f 


.u 


-11 


.54 


hV 


IZQ 


ZZO 


140 


40 


7? 


Z3i> 


3r<l3 


5^ 


n* 


-7' 


h 


fAi 


315 


ZZ5 


136 


4S 


^1 


2,^ 


4-MI 


5S 


'17 


.U 


Ucf 


hSif 


310 


Z80 


130 


SO 


?^ 


Z4& 


4 JO 


53SL 


.&z 


^7 


1.41 


h74 


30s 


Z35 


IZ5 


55 


MS 


2,0^ 


4rf^ 


&74 


37 


.S 


I.] 3 


z. 


3 00 


ZAO 


IZO 


to 


m 


z,ot 


4jtd 


5,15 


^' 


^41 


Z.lA- 


Z.3i 


Zt^S 


Z45 


US 


^3 


a2 


I.CIZ 


4.3^ 


5fib 


'?4 


.34 


^7^ 


z.<iz 


ZcfO 


Z50 


110 


10 


^3^ 


183 


4,4s 


4^<n 


•?7 


.2^ 


3.7s 


B^ 


ZB5 


zss 


106 


7^ 


m 


tjs 


4M 


4^ 


.f8 


''T 


S^] 


5.7^ 


zeo 


z(,o 


too 


&o 


m 


tM5 


AU 


4^0 


•^? 


^? 


rU3 


UAt 


z-js 


Zh5 


95 


&5 


f-si 


h&l 


4'ni 


^m. 


^ 


a 


HONE 


NOHE 


ZfO 


ifo 


u^ 


SO 



155 



RADIANS 


CYCLES 


D£GRBES 


b^ZQ3 


f 


3f>o 


iz.sif^ 


z 


1Z0 


l6,dS0 


3 


1 o&o 


2$J33 


4 


/440 


3/.4/(^ 


5 


/ doo 


37-^9 


^ 


Zl ^0- 


43.qez. 


7 


Z5^0 


S0,Z^5 


6 


a 660 


5^,54(i 


^ 


5&40 


U'^3Z 


/O 


-^li^oo , 



Natural or 6-Logarithms, Bose: Z,lld-*- \ 


N 


i. 


■Kl— /3 ^ / — / fcf 


N 


u 






N- C L~lo^^N - 






10 


2.303 


./ 


rb<fj-io 


too 


4.^05 


(s/ in Kie margin 


.OJ 


5.3Cj5-lO 


IO0Q 


^.<7C?S 


L in the table 


.001 


3.O9Z-I0 


N 


-O 


.1 


.2. 


,3 


.4 


.6 


.h 


.7 


.8 


.9 


I. 


Q 


0,0^6 


0J8Z 


o.z^z 


0.33^ 


0,405 


OAIO 


0.531 


0,588 


oMZ 


z. 


A^f3 


0.74Z 


oqea 


0.633 


0.875 


0,<flh 


o,<jbI> 


C,ff3 


1.030 


I'OU 


3. 


'•<'?'? 


1.131 


Ui>3 


1.1 ff A 


1.ZM 


J.Z53 


i.zdi 


1.305 


1335 


l.3i>t 


4. 


/.aa^ 


i.Ht 


1435 


l'45Cf 


l.4dl 


1.604 


/.J2^ 


I.Sid 


ISlf 


is&i 


5, 


^^^f 


Ut<i 


Uh 


I Us 


U6(, 


1,105 


1.7 Z2 


1.740 


1.75-6 


1.775 


L 


i7f-^ 


lece 


ms 


1.84 i 


l.6Bb 


I.81Z 


hae't 


I.^Ol 


A 9/7 


t.t^3Z 


% 


/f¥^ 


vf^^ 


'-m 


J.9&6 


Z.ooi 


;i.ois 


Z.0Z6 


2J>4\ 


2.054- 


^Pil 


A 


z.07cf 


2.^^ 


I.IH 


Z.lU 


z.i%i 


Z.I40 


£.151 


i.fU 


2.175 


zM 


% 


mi. 


^.2<« 


;-2i<? 


K->40, 


s^l 


Z.ZSi\zH^ 


Z.Z7? 


i.9.8Z 


^.Z<^i 



15^ 

THB PlACE OF FrESHMAH CALtULUS 
IN THE CUfiRICOLUtA 

Af Tufts Collbqb Me prescribed maZ-he-- 
mahcs in fh€ Bhijjneerihj Depart merit- con- 
sists of the Mlot^i'r?^ courses; 

Course 1, C^mpuiah^. Ctse qF tnyonom^tric 
functions^ lyefnthm^, sti'de-rulexofdicoits ancf 
combincrhoh numbers, Pl&ne fnfurfgles, 
Course ^, Algebraic Oifid ^rQph^fii methods, 
dmult^peei/s and quad rot tit eguatfom^ Co- 
ordihates^ qrophs^ straight Une\ cird^s, and 
corfi'z se€t/(ms, locus probfem$, 

ZndT^nMj t^r?£SHMAN'^0AT^ 
Course 3. 771 IS text-Joofh is txpresslj^ in- 
t^hded far thi^ cour&z^ 

IsK TE/^p% 3opH0 Mcm^ re: A R 
Course 4. The more admnced part& of /t'^'^fi' 
nomQtnf and e/em&ntcirij ecffcu(i<^ ffenew^dritt 

Znd. TEffM^ SaPHOMOFtE YeAT^ 

Courses, 7 tire % dime h si on a I yvorh in tn'^o- 
nometrj^, anaJj^h'tQi qtomehy and calculus, 
in troxiuetjoh to dirteren tiat e<fUOi tions , 

Ji/NI0H AND S^r^/0/f V^APfS 
Privilege of e/ecttnj hC^her mathe^ 
ma he s f /yen in the Co/fej^ of Letters, 



157 

While this h%t- book has been designed h 
tit Q eertQ/'n ccjrnculUM, . //^ grran^^- 
memt fs such as^ ho permif' ih usein_ 
a variety of corses, ft majf be hhen up 
slinultQh eousfy with > h^ifono m etr f^ & Hd^^ 
Qnalf^ddQl ^^ometrci w /-he hrsf- Aez-w, 
since no hrctnscen dental's oippeoi/^ m the 
f-frst eo pct^esCsave in /f'/2 3 /q^s). It 
YYictCf be used }n reriepv in the iater 
*fe<irs smce ft cfea/s so copiouslfj vr'tth 
the important oipfol teat ions of the calcu- 
lus ]h a/7 en^ineeri'n^ or ^sc/ent'/t/c 
course, For such a purpose the chhf omis- 
^fon that wftl be noted is that ot centers 
of ^rayihj and moment's ot in ertict^qnct 
muttlptt, integration. 

The nu/nber of concrete problems is 
lar^e, f<f^ out of the 417^ oind Includes 
motnjf problems of ritdl tnoportance in 
the studjf of pHtfSJCS, and apt to hare 
their mathemattcal features fnsuffieientijj 
considered in fhe Science courses'. 

This hext makes no pretension to com- 
pleteness, btit i^ intended to preee^de a 
caloulas treatfnj ai grreaterranje of topics, 
On/jf such topics are here presented as ^ 
Sophomore tnaineer is h'helj/ to reQuir^, 



m 



Specially Named Rates 



if /s c/ef/he^ as fhe raid ^ af ihcrease 
of S /n un/fs t?f S pfr un/f of /hcr^ase 
of T b(/ an e^c/af/h^ ^f fAe -Porm', 



/mfances /h 
Speecf = 

Acce/eraf^hn- 
Foir/er «= 

Curreni' — 
^ Slope = 

Force ^ - 
Cross- section^ 

Pressure - 

Densitc/ — 
Specific heaf'=' 



ffris fert- are \ 
A {4 r static e)-r-A (//me) 
4 Cspee(/) -f ^^/Wj 
A (t^or/c) -r'A(f/'rie) 
A[ro/ome) -.A[f7mcJ 
4 (rise } -T 4 (run) 
A (area) 
A \frork) 



A X-fa^e) 
4 (force) 
Afmass^ 



-rA (basj) 
-r-AL^islonce) 
A (rohme) -r A [len^fh) 



A (c/i/fc/^ce) 
A (area) 



^s) -^ Afro/ o me) 
A [heafj -^A(f^mpera/Pr^ 



The (/mts rr/;/c/? preceate c?/7^ -^/JoyrAeperfn 
'ffie name of m€ c/n/f of agrafe i^hose i/wf is 
not spec/af/y na/pfeaf scffesf m oeFmtr\oNof 
ffffif rat^: fofisi/e sfrenfif? fs measore^/h/h, 
pfrjf'/kj Afnoe /s a^ef/neal asA^/ve)rA(ar&^ 



JAN IS m^ 





■ 







LIBRARY OF 




003 527 028 



